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Alain Connes

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Riemann–Roch for the ring $\mathbb{Z}$

Vicente Lorenzo

On the group of automorphisms of Horikawa surfaces

Anderson Porto

Igor Lima

Comptes Rendus
Mathématique
Anderson Porto and Igor Lima
On Pro-p Cappitt Groups with finite exponent
Volume 362 (2024), p. 287-292
Online since: 2 May 2024
https://doi.org/10.5802/crmath.562
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 287-292
https://doi.org/10.5802/crmath.562
Research article / Article de recherche
Group theory / Théorie des groupes
On Pro-p Cappitt Groups with finite exponent
Anderson Porto ,a,b and Igor Lima ∗,a,b
a Instituto de Ciência e Tecnologia - ICT, Universidade Federal dos Vales do Jequitinhonha e Mucuri,
Diamantina - MG, 39100-000 Brazil
b Departamento de Matemática, Universidade de Brasília, Brasilia-DF, 70910-900 Brazil
E-mails: ander.porto@ict.ufvjm.edu.br, igor.matematico@gmail.com
To Pavel Zalesskii on his 60th birthday
Abstract. A pro-p Cappitt group is a pro-p group G such that S̃(G) = 〈L ⩽c G |L ⋪G〉 is a proper subgroup (i.e.
S̃(G) ̸= G). In this paper we prove that non-abelian pro-p Cappitt groups whose torsion subgroup is closed
and it has finite exponent. This result is a natural continuation of main result of the first author [7]. We also
prove that in a pro-p Cappitt group its subgroup commutator is a procyclic central subgroup. Finally we show
that pro-2 Cappitt groups of exponent 4 are pro-2 Dedekind groups. These results are pro-p versions of the
generalized Dedekind groups studied by Cappitt (see Theorem 1 and Lemma 7 in [1]).
Keywords. Generalized Dedekind groups, pro-p Cappitt groups, torsion groups.
2020 Mathematics Subject Classification. 20E34, 20E18.
Funding. The first author was partially supported by FAPDF, Brazil. The second author was partially sup-
ported by DPI/UnB, FEMAT Proc. 054/2022, FAPDF, Brazil.
Manuscript received 26 June 2023, revised and accepted 2 September 2023.
1. Introduction
Profinite groups are totally disconnected compact Hausdorff topological groups. Such groups
can be seen as projective limits of finite groups (see Ribes–Zalesskii [9] or Wilson [12]).
We recall that a totally disconnected compact Hausdorff group G is called a profinite torsion
group (or periodic) if all of its elements are of finite order.
One motivation for studying these groups is the following open question due Hewitt–Ross (see
Open Question 4.8.5b in [9]): Is a profinite torsion group necessarily of finite exponent?
In celebrated paper [13], Zel′manov proves that a finitely generated pro-p torsion group is
finite. Therefore the question above is open for infinitely generated torsion groups.
In [3] and [4], Herfort proves that a profinite group G whose order is divisible by infinitely many
different primes has a procyclic subgroup with the same property. Otherwise if G is a profinite
torsion group then the order of G is divisible by only finitely many distinct primes.
As main result in this paper we prove that a infinitely generated non-abelian pro-p Cappitt
group has finite exponent. This result is the profinite version of second part of Theorem 1 in [1].
In [7] the first author of this paper proves the first part of the our main result (see Theorem A).
Also we obtained other related results of independent interest (see Theorem B).
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
288 Anderson Porto and Igor Lima
2. Preliminaries
The standard notation is in accordance with [7], [9] and [12].
We recall that an abstract group (finite or infinite) is called a Dedekind group if every subgroup
is normal. As in [1], for any group G , we define S(G) to be the subgroup of G generated by
all the subgroups which are not normal in G , i.e. S(G) = 〈N |N ≤ G and N ̸⊴ G〉. Note that G
is a Dedekind group if and only if S(G) = {1}. Also S(G) is a characteristic subgroup of G and
the quotient group of G by S(G) is a Dedekind group. A group is to be a Cappitt group if it
satisfies S(G) ̸= G , these groups are generalizations of Dedekind groups (see for example Kappe
and Reboli [5] or Cappitt [1]).
In the profinite version, as in [7] and [8], a profinite Dedekind group will be a profinite group
in which every closed subgroup is normal. A profinite non-abelian Dedekind group will be called
a profinite Hamiltonian group.
Remark 1. If each procyclic subgroup of a profinite group G is normal, then G is a profinite
Dedekind group (see Remark 1, p. 90-91 in [8]).
A. Porto and V. Bessa [8] classified these groups as follow
Theorem 2. A profinite group G is Dedekind if, and only if, G is abelian or there exists a finite set
of odd primes J and a natural number e such that
G ∼=Q8 × Ẽ ×
∏
p∈J
(
e∏
i=1
( ∏
m(i ,p)
Cp i
))
,
where Q8 is the quaternion group of order 8, Ẽ is an elementary abelian pro-2 group and each
m(i , p) is a cardinal number. In particular, if G is a profinite Hamiltonian group then G has finite
exponent.
Define G ′ as the topological closure in G of the abstract commutator subgroup [G ,G] =
〈[g ,h] |g ,h ∈ G〉, where [g ,h] = g−1h−1g h is the commutator of the elements g ,h ∈ G . The
abelianization of group G will be denoted by Gab = G/G ′ = G/[G ,G]. If G is a profinite group,
Gab is an abelian profinite group.
Definition. A profinite Cappitt group is a profinite group G such that S̃(G) = 〈L ⩽c G | L ⋪G〉 is a
proper subgroup of G.
This profinite version was defined firstly in [7]. For finite groups we have that S̃(G) = S(G).
Using Theorem 2, Porto gave the following characterization of profinite Cappitt groups.
Theorem 3. Let G be a profinite group satisfying {1} ̸= S̃(G) ̸=G . Then G can be expressed as direct
product H̃ × K̃ , where H̃ is a pro-p Cappitt group for some prime p and K̃ is a profinite Dedekind
group that does not contain elements of order p. Such G is nilpotent profinite of class at most 2.
Denote by Zp the p-adic integers. Consider |z| the order of an element z of a group G . For a
group G we denote tor(G) = {x ∈ G | |x| <∞} the subset of the elements of torsion of G . As in the
abstract case (see 16.2.7 in [6]), when G is a nilpotent pro-p group, then tor(G) is a subgroup of G
(not necessarily closed). A main result of this paper is the following
Theorem A. Let G be a non-abelian pro-p Cappitt group. Then G ′ is a procyclic central subgroup.
Moreover, if tor(G)⩽c G then G has finite exponent.
Remark 4. The Theorem A has immediate proof when G is a pro-2 Hamiltonian group since
G ′ = C2 is central because G has nilpotent class 2 (see Corollary 1 in [7]), and also G has finite
exponent by Theorem 2. The condition to be non-abelian is necessary in the Theorem A sinceZp
is an abelian non-periodic group satisfying all hypothesis.
Anderson Porto and Igor Lima 289
Now we present some preliminaries results that will are used throughout in the proof of the
main theorem.
Lemma 5. Let G = lim←−−i∈I {Gi ,φi j } where {Gi ,φi j ,I } is a surjective inverse system of finite groups
(with discrete topology). Then
S̃(G) = lim←−−
i∈I
{S(Gi ),φi j |S(Gi ) },
where {S(Gi ),φi j |S(Gi ) ,I } is the associated surjective inverse system.
Proof. See Proposition 1 in [7]. □
The following facts are direct consequences of the definition of subgroups S(G),S(L), S(G/J )
and the Correspondence Theorem. In the case of abstract groups, these can be found in remark
on page 312 in [1] or Lemma 2.2 in [5].
Remark 6. If G is an abstract group with S(G) ̸=G and J is a normal subgroup of G contained in
S(G), then S(G/J ) ̸=G/J , furthermore if L is a subgroup of G containing elements of G \S(G) then
S(L) ̸= L.
Analogously to the abstract case, similar results can be shown in the category of profinite
groups, as follows.
Remark 7. If G is a profinite Cappitt group (i.e. S̃(G) ̸=G) and J is a closed normal subgroup of
G contained in S̃(G), then S̃(G/J ) ̸= G/J , furthermore if L is a closed subgroup of G containing
elements of G \ S̃(G) then S̃(L) ̸= L.
The following result is similar to Lemma 1 in [7], we will repeat the same argument for the
convenience of the reader.
Lemma 8. Let G be as in the Lemma 5. If G ̸= S̃(G) then Gi ̸= S(Gi ), ∀ i ∈I .
Proof. Suppose that there is a Gi such that S̃(Gi ) = S(Gi ) = Gi . Denote by φi : G ↠ Gi the
continuous canonical projection of the inverse limit described in Lemma 5. By assumption Gi
is generated by all its non-normal subgroups. Let L be one of these subgroups and consider
Li its inverse image by φi , clearly Li is a non-normal closed subgroup of G containing ker(φi ).
Therefore ker(φi ) ⩽ S̃(G) and by Remark 7 have G/ker(φi ) ̸= S̃(G/ker(φi )) = S(G/ker(φi )), so
|S(Gi )| < |Gi | <∞, a contradiction. □
In accordance with the notations of the previous lemmas we obtain the following.
Lemma 9. Let G be a pro-p Cappitt group. Then G ′ is a procyclic central subgroup.
Proof. Since Cappitt groups are nilpotent of class at most 2 (see Corollary 1 in [7]) we have
G ′ = [G ,G] ⩽ Z (G). Consider G = lim←−−i∈I {Gi ,φi j } where {Gi ,φi j ,I } is a surjective inverse system
of finite p-groups (with discrete topology). From Lemma 8 it follows that Gi ̸= S(Gi ), ∀ i ∈ I .
Now, if any Gi is abelian we have (Gi ) ′ = {1}. On the other side, if Gi is a non-abelian p-group then
from Theorem 1 in [1], we have (Gi )′ is a finite cyclic p-group. Note that φi (G ′) = (Gi ) ′,∀ i ∈ I ,
hence from Corollary 1.1.8 in [9] we have G ′ = lim←−−i∈I {(Gi )
′,φi j } whence it follows that G ′ is
procyclic. □
The preceding result is a generalization of the Corollary on page 314 in Cappitt [1]. Kappe and
Reboli show that the proof given by Cappitt was incorrect and give a new proof for this fact, this
is the content of Theorem 4.2 in Kappe and Reboli [5].
Remark 10. If G is a non-abelian pro-p Cappitt group, then G is nilpotent of class 2 (see Corol-
lary 1 in [7]). In this case G ′ ⩽ Z (G) and therefore [[x, y], z] = 1, ∀ x, y, z ∈G . By Lemma 5.42(i) in
Rotman (see [11]), we have that [xn , y] = [x, yn] = [x, y]n , ∀ n ∈Z.
290 Anderson Porto and Igor Lima
Lemma 11. Let G be a non-abelian pro-p Cappitt group. If a ∈G \ S̃(G) has infinite order, then a
is central.
Proof. Let a be an element in G \ S̃(G) of infinite order, hence |a| = p∞ and 〈a〉 ∼= Zp . If a is not
central in G then there is g ∈ G such that [a, g ] ̸= 1, moreover g−1ag ∈ 〈a〉 because 〈a〉◁G . By
Remark 10 we have g−1ag a−1 = [g , a−1] = [g , a]−1 = [a, g ] ∈ 〈a〉. Since Zp is a free pro-p group
on {1}, there is a unique continuous epimorphism ζ :Zp →〈a〉 such that ζ(1) = a, therefore each
element of 〈a〉 can be written as aλ = ζ(λ) for some λ ∈ Zp (see similar notation in Section 4.1
on [9]). Since 〈a〉 is infinite it follows that ζ is an isomorphism (see Proposition 2.7.1 in [9]).
In particular 1 ̸= [a, g ] = aδ for some δ ∈ Zp . Let (ni )i∈N be a sequence of integers converging
to δ on Zp , in other words choose the ni such that limi→∞ ni = δ (note that 〈a〉 = Zp is a
complete metric space and Z = Zp ). Adapting the argument from Lemma 4.1.1 in [9] to Zp we
have limi→∞ [a, g ]ni = [a, g ]δ. By properties of limits of sequences in metric spaces together with
Remark 10 we obtain
[a, g ]δ = lim
i→∞
[ani , g ] = lim
i→∞
(ani )−1 · lim
i→∞
g−1 · lim
i→∞
ani · lim
i→∞
g = [aδ, g ].
As G ′ is central (see Remark 10) we have [a, g ] = aδ ∈ Z (G). So [[a, g ], g ] = [aδ, g ] = [a, g ]δ = 1,
which implies that a2·δ = ζ(2 ·δ) = 1, a contradiction because Ker(ζ) = {0}. Therefore a is central
in G . □
Proposition 12. Let G be a non-abelian pro-p Cappitt group and tor(G) is a closed subgroup of
G. Then G is a periodic group.
Proof. Suppose that G contains elements of infinite order. Since G is nilpotent whose class
is two (see Corollary 1 in [7]), it is generated by these elements, not all of which lie in S̃(G)
otherwise we would have G = 〈G \ S̃(G)〉= 〈tor(G)〉 = tor(G), a contradiction. Let a be an element
in G \ S̃(G) of infinite order. From Lemma 11 a is central. Now if w is a non-central element
of G \ S̃(G) then aw, a2w are non-central elements of infinite order, so both lie in S̃(G). Thus
a2w w−1a−1 = a ∈ S̃(G), contradicting the choice of a. Therefore every element of G \ S̃(G) is
central and thus G is abelian since G \ S̃(G) generates G , a contradiction. □
Example. Note that the condition S̃(G) ̸= G in the Proposition 12 is necessary, for instance
G = (∏r∈N Zr )×Q8 is a non-abelian pro-2 non-periodic group with tor(G) = Q8 and Zr ∼= Z2,
∀ r ∈ N. We show that S̃(G) = G . Fix n ∈ N, indeed, it is sufficient we consider the following
families of procyclic subgroups of G isomorphic to Z2, that we will denote by
In = 〈(0, . . . ,0,1n ,0, . . . ,0, i )〉,
Jn = 〈(0, . . . ,0,1n ,0, . . . ,0, j )〉,
Kn = 〈(0, . . . ,0,1n ,0, . . . ,0,k)〉,
where 1n := (0, . . . ,0,1n ,0, . . . ,1) is the generator of Zn ∼=Z2. Observe that
(0, . . . ,0,1n ,0, . . . ,0, i )
j = (0, . . . ,0,1n ,0, . . . ,0,−i ) ̸∈ In ,
so inductively In is not a normal subgroup of G . Similarly, it can be shown also that Jn and
Kn are not normal subgroups of G for each n ∈ N. Therefore by the definition of S̃(G) we have
that In , Jn and Kn are contained in S̃(G). Now consider the following elements of S̃(G), xn =
(0, . . . ,0,1n ,0, . . . , i ) and yn = (0, . . . ,0,1n ,0, . . . ,0, j ). Note that xn · y−1n = (0, . . . ,0,−k) := −k, so k ∈
S̃(G). Similarly i , j ∈ S̃(G) and this implies that Q8 is contained in S̃(G). Since (0, . . . ,0,1n ,0, . . . ,0,k)
and (0, . . . ,0,−k) are in S̃(G) we have 1n := (0, . . . ,1n ,0, . . . ,0,1) belongs to S̃(G), and therefore
S̃(G) =G .
Anderson Porto and Igor Lima 291
3. Proof of the Theorem A
We are now ready to prove a main result of this paper.
Proof. The fact that G ′ is a procyclic central subgroup follows from Lemma 9. If G is Dedekind
(i.e. pro-2 Hamiltonian group) the result follows immediately from Theorem 2 and Remark 4.
From now on we will consider that S̃(G) ̸= {1}. Since tor(G) is closed in G , from Proposition 12
follows that G is a pro-p torsion group. If G is finitely generated as pro-p group then from
celebrated Zel′manov Theorem (see Theorem 4.8.5c in [9]) we have that G is a finite p-group
and the result follows. Therefore we can suppose that G is an infinitely generated pro-p and {1} ̸=
S̃(G) ̸=G . Also since G is a torsion group and G ′ is procyclic, it follows from Proposition 2.7.1 [9]
that G ′ =Cpn for some n ∈N fixed.
Suppose by contradiction that G has no finite exponent. So there are p-elements of G , say,
x1, x2, . . . , xk , . . . , (k ∈N), such that |xk |→∞ for k →∞ (elements of unlimited order).
Let Gab =G/[G ,G] be the abelianization of G . It is straightforward prove that Gab is an abelian
pro-p torsion group because G is a torsion group. From Theorem 4.3.8 in [9] we obtain that
Gab = ∏
m(1)
Cp ×
∏
m(2)
Cp2 × . . .×
∏
m(e)
Cpe ,
where each m(i ) is a cardinal number, e is some natural number, with at least one m(i ) infinite
because Gab is infinitely generated.
For each g ∈ G we have (gG ′)pe = G ′ since pe is the finite exponent of Gab . Therefore for all
k ∈Nwe have
G ′ = (xkG ′)p
e = xpek G ′ =⇒ x
pe
k ∈G ′ =Cpn .
Thus we obtain
(xk
pe )p
n = xpe+nk = 1.
Therefore the order of each xk is limited by p
e+n , ∀ k ∈ N, a contradiction. We conclude that G
has finite exponent. □
4. Pro-2 Cappitt groups of exponent 4
The following result shows that the only pro-2 Cappitt groups of exponent 4 are pro-2 Dedekind
groups.
Theorem B. If G is a pro-2 group of exponent 4 then either S̃(G) = {1} or S̃(G) =G.
Proof. Suppose that {1} ̸= S̃(G) ̸= G . It is clear that not every procyclic subgroup of G is normal,
otherwise S̃(G) = {1} (see Remark 1). Then consider J = 〈x〉 a non-normal procyclic subgroup of G .
Let H = 〈x, y, z〉⩽c G such that y ̸∈ NG (J ) and z ̸∈ S̃(G). By construction and Remark 6 we have that
H is a 3-generated group of exponent 4 with {1} ̸= S̃(H) ̸= H and 〈z〉⊴c H . Since H has exponent
4, we have that H is a finitely generated pro-2 torsion group, it follows from celebrated Zel′manov
Theorem that H is a finite non-abelian 2-group (see Theorem 4.8.5c in [9]). From Corollary 1 in
Porto [7] and Lemma 9 we have that G and H are nilpotent groups of class 2 with H ′ being cyclic,
so H ′ ∼= C2 or H ′ ∼= C4. Take a minimal set of generators of H chosen to lie outside S̃(H) = S(H),
say, {h1,h2, . . . ,hl }, where l ≥ 2. Note that H ′ =
〈
[hi ,h j ] | i , j ∈ {1,2, . . . , l }, i ̸= j
〉
(see p. 129 in [10]).
Let Hi = 〈hi 〉 for each i ∈ {1,2, . . . , l }. Since hi ̸∈ S(H) we have Hi ⊴ H and so [hi ,h j ] ∈ Hi ∩H j for
each i , j ∈ {1,2, . . . , l }. Note that [hi ,h j ] belongs to Hm for every i , j ,m ∈ {1,2, . . . , l }, so H ′ ⩽ Hm
for each m ∈ {1,2, . . . , l }. If H ′ ∼=C4 we have that H ′ = H1 = H2 = ·· · = Hm , a contradiction since H
is not abelian. Therefore H ′ has order 2. Note that H is a finite 2-group and it is 3-generated with
exponent 4, so it is well-know that its order is at most 64. A GAP computation using [2] shows that
H is either Dedekind or S̃(H) = S(H) = H , a contradiction. □
292 Anderson Porto and Igor Lima
For example, a GAP [2] check yields, the SmallGroup(16, 6), SmallGroup(27, 4) and Small-
Group(64, 28) are finite Cappitt groups with commutator subgroup C2, C3 and C4, respectively.
In Corollary 4.4 in [5], Kappe and Reboli prove that if G is a p-group Cappitt, then G ′ is a finite
cyclic p-group. Finally, in view of this result, example above, Lemma 9 and Theorem A, we can
pose the following questions.
5. Open questions about profinite Cappitt groups
(1) Are there non-abelian non-periodic pro-p Cappitt groups? If G is not torsion, is G
virtually p-adic?
(2) If G is a pro-p group satisfying {1} ̸= S̃(G) ̸=G , do we always have that G ′ ∼=Cpn for some
n ∈N?
Acknowledgment
We thank John MacQuarrie for the careful reading and suggestions.
References
[1] D. Cappitt, “Generalized Dedekind groups”, J. Algebra 17 (1971), p. 310-316.
[2] The GAP Group, “GAP — Groups, Algorithms, and Programming, Version 4.10.2”, 2019, http://www.gap-system.org.
[3] W. N. Herfort, “Compact torsion groups and finite exponent”, Arch. Math. 33 (1980), p. 404-410.
[4] ——— , “An arithmetic property of profinite groups”, Manuscr. Math. 37 (1982), p. 11-17.
[5] L.-C. Kappe, D. M. Reboli, “On the structure of generalized Hamiltonian groups”, Arch. Math. 75 (2000), no. 5, p. 328-
337.
[6] M. I. Kargapolov, J. I. Merzljakov, Fundamentals of the Theory of Groups, Graduate Texts in Mathematics, vol. 62,
Springer, 1979.
[7] A. L. P. Porto, “Profinite Cappitt groups”, Quaest. Math. 44 (2021), no. 3, p. 307-311.
[8] A. L. P. Porto, V. R. de Bessa, “Profinite Dedekind groups”, Far East J. Math. Sci. 126 (2020), no. 1, p. 89-97.
[9] L. Ribes, P. Zalesskii, Profinite groups, 2nd ed., Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, vol. 40,
Springer, 2010.
[10] D. J. S. Robinson, A course in the Theory of Groups, Graduate Texts in Mathematics, vol. 80, Springer, 1996.
[11] J. J. Rotman, An introduction to the theory of Groups, 4th ed., Graduate Texts in Mathematics, vol. 148, Springer, 1995.
[12] J. S. Wilson, Profinite Groups, London Mathematical Society Monographs. New Series, Clarendon Press, 1998.
[13] E. I. Zel′manov, “On periodic compact groups”, Isr. J. Math. 77 (1992), no. 1-2, p. 83-95.


On Pro-$p$ Cappitt Groups with finite exponent

Boris Kunyavskiĭ

Tori and surfaces violating a local-to-global principle for rationality

Mohamed Jleli

Bessem Samet

On the critical behavior for a Sobolev-type inequality with Hardy potential

Lucio Boccardo

Juan Casado-Diaz

Luigi Orsina

Comptes Rendus
Mathématique
Lucio Boccardo, Juan Casado-Diaz and Luigi Orsina
Dirichlet problems with skew-symmetric drift terms
Volume 362 (2024), p. 301-306
Online since: 2 May 2024
https://doi.org/10.5802/crmath.564
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 301-306
https://doi.org/10.5802/crmath.564
Research article / Article de recherche
Partial differential equations / Équations aux dérivées partielles
Dirichlet problems with skew-symmetric drift
terms
Problèmes de Dirichlet avec des termes de drift
asymétriques
Lucio Boccardo a, Juan Casado-Diaz b and Luigi Orsina ∗,c
a Istituto Lombardo & Sapienza Università di Roma, Italy
b Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, Spain
c Dipartimento di Matematica, Sapienza Università di Roma, Italy
E-mails: boccardo@mat.uniroma1.it, jcasadod@us.es, orsina@mat.uniroma1.it
Abstract. We prove existence of finite energy solutions for a linear Dirichlet problem with a drift and a
convection term of the form A E(x)∇u +div(u E(x)), with A > 0 and E in (Lr (Ω))N . The result is obtained
using a nonlinear function of u as test function, in order to “cancel” this term.
Résumé. Nous prouvons l’existence de solutions d’énergie finie pour un problème de Dirichlet linéaire avec
un terme de la forme A E(x)∇u + div(u E(x)), où A > 0 et E est dans (Lr (Ω))N . Le résultat est obtenu en
utilisant une fonction non linéaire de u comme fonction test, afin d’“annuler” ce terme.
Keywords. Singular drift, Dirichlet problems, nonlinear test functions.
Manuscript received 4 August 2023, revised 4 September 2023, accepted 6 September 2023.
1. Introduction
In [8] and [9] is studied the Dirichlet problem
u ∈W 1,20 (Ω) : −∆u +E(x) ·∇u +div(u E(x)) = f (x),
where
Ω is a bounded domain of RN , and the drift E belongs to (L2(Ω))N ,
and
f (x) ∈ L 2NN+2 (Ω).
Remark that if u belongs to W 1,20 (Ω), then the terms E(x) ·∇u and div(u E(x)) just belong to L1(Ω)
and W −1,
N
N−1 (Ω) (or W −1,p (Ω), ∀ p < 2 if N = 2) respectively. The key point for the existence of
finite energy solutions is that the map
P : u → E(x) ·∇u +div(u E(x))
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
302 Lucio Boccardo, Juan Casado-Diaz and Luigi Orsina
is skew-symmetric and thus, it satisfies the “cancellation” property
〈Pu,u〉 = 0, (1)
in the sense of distributions, if u is smooth enough.
The present paper deals with a more general framework, given by
u ∈W 1,q0 (Ω) : −div(M(x)∇u)+ A E(x) ·∇u +div(u E(x)) = f (x), (2)
with
A > 0, (3)
M :Ω→RN 2 a measurable matrix such that there exist α, β> 0, satisfying
α|ξ|2 ≤ M(x)ξ ·ξ, |M(x)| ≤β, a.e. inΩ, ∀ ξ ∈RN , (4)
E ∈ (Lr (Ω))N :

r = 2 if A ≥ 1,
r = N (A+1)A(N−1)+1 if 0 < A < 1, N ≥ 3,
2 < r < 1+ 1A if 0 < A < 1, N = 2,
(5)
and
f ∈ Lm(Ω), m > 1 if N = 2, m = N (A+1)
N +2A if N ≥ 3. (6)
Note that r ≥ 2, so that r ′ ≤ 2; and also r ′ > 1.
We are going to prove the existence of a distributional solution u of (2) in the Sobolev space
W 1,r
′
0 (Ω), with r
′ = rr−1 , the Hölder conjugate exponent of r . In particular, u is in W 1,20 (Ω) if A ≥ 1.
The proof of the result is based on the use of a nonlinear function of u as test function in (2),
in such way that a cancellation property similar to (1) still holds.
We complete this introduction with some references about the existence of solutions for linear
elliptic equations with a first order term whose coefficients have poor summability.
For a measurable matrix function M :Ω→RN 2 , which satisfies (4),
E ,F ∈
{
(LN (Ω))N if N > 2,
(Lp (Ω))N , p > 2 if N = 2, a ∈
{
L
N
2 (Ω) if N > 2,
Lq (Ω), q > 1 if N = 2, (7)
such that
−div(E(x))+a(x) ≥ 0 inΩ, (8)
it has been proved in [11, 12] that the weak maximum principle holds for the equation
L u = f in Ω,
with
L u =−div(M(x)∇u −u E(x))+F (x) ·∇u +a(x)u. (9)
Thus, the corresponding Dirichlet problem has at most one solution in W 1,20 (Ω), for every f ∈
W −1,2(Ω). By the Fredholm theory, for every f ∈ W −1,2(Ω) there exists a unique solution u for
problems
u ∈W 1,20 (Ω), L (u) = f in Ω, u ∈W 1,20 (Ω), L ∗(u) = f in Ω, (10)
where
L ∗(u) =−div(M(x)t∇u +u F (x))−E(x) ·∇u +a(x)u,
is the adjoint operator of L . We emphasize that these problems are not coercive.
If E or F do not satisfy (7), the operators u 7→ −div(u E(x)) or u 7→ F (x)·∇u do not apply W 1,20 (Ω)
into W −1,2(Ω), but some existence results have still been proved.
In [3] and [4], are considered the cases F = 0 and E = 0 respectively, and a Stampacchia–
Caldéron–Zygmund theory for finite or infinite energy solutions (depending on the summability
of f (x)) is proved. For example, if E ∈ (L2(Ω))N , F = 0, f ∈ L1(Ω), it is proved the existence of a
solution for the problem
L u = f in Ω, u = 0 on ∂Ω, (11)
Lucio Boccardo, Juan Casado-Diaz and Luigi Orsina 303
in a very weak sense: the solution u is such that log(1+|u|) belongs to W 1,20 (Ω) and is an “entropy
solution” of the equation (for the theory of entropy, or renormalized, solutions, see e.g. [1], [7],
[10]) in the sense that:∫
Ω
[
M(x)∇u ·∇Tk (u −ϕ)−u E(x) ·∇Tk (u −ϕ)
]
≤
∫
f (x)Tk (u −ϕ),
for every ϕ in W 1,20 (Ω)∩L∞(Ω), and for every k > 0. Here
Tk (s) =

−k if s <−k,
s if −k ≤ s ≤ k,
k if s > k,
(12)
is the usual truncation function at levels ±k. In [6], adding a zero order term greater than
a positive constant, it has been proved that the above function u is also in L1(Ω). A duality
argument then shows that problem (11) has a solution in W 1,20 (Ω)∩L∞(Ω) if E = 0, F is in (L2(Ω))N
and a is greater than a positive constant.
2. Proof of the main result
We devote this section to the proof of an existence result for problem (2). It will be based on the
introduction of an approximate problem and then the use of a nonlinear test function, which will
provide the estimates needed to pass to the limit.
Theorem 1. Assume that M :Ω→RN 2 is a measurable matrix which satisfies (4), that A > 0, and
that E belongs to (Lr (Ω))N , with r defined by (5). Then, for every f ∈ Lm(Ω), with m given by (6),
there exists a distributional solution u of (2) in W 1,r
′
0 (Ω). Moreover, u satisfies
|u| A−12 u ∈W 1,20 (Ω), u ∈
{
L
N (A+1)
N−2 (Ω) if N ≥ 3,
Lp (Ω), ∀ p ∈ [1,∞) if N = 2. (13)
Remark 2. We point out a regularizing effect of the problem: for A ≥ 1, u belongs to W 1,20 (Ω),
even if the term A E(x) ·∇u only belongs to L1(Ω).
Remark 3. Defining for µ,λ > 0, Ê(x) = λE(x) and A = µλ, we deduce from Theorem 1 the
existence of a distributional solution for problem
u ∈W 1,r ′0 (Ω) : −div(M(x)∇u)+λE(x) ·∇u +µdiv(u E(x)) = f (x).
Proof of Theorem 1. Let n in N: the starting point is the nonlinear Dirichlet problem: un ∈
W 1,20 (Ω) :∫
Ω
M(x)∇un ·∇v −
∫
Ω
un
1+ 1n |un |
En(x)
1+ 1n |∇un |
·∇v
+ A
∫
Ω
1
1+ 1n |un |
En(x)
1+ 1n |∇un |
·∇un v =
∫
Ω
fn(x) v, ∀ v ∈W 1,20 (Ω), (14)
where
En(x) = E(x)
1+ 1n |E(x)|
, fn(x) = f (x)
1+ 1n | f (x)|
.
The existence of un is a consequence of the use of the Schauder fixed point theorem (see also [2]),
since all the terms are bounded. Note that un is the solution of the Dirichlet problem
un ∈W 1,20 (Ω) : −div(M(x)∇un) =−div(Gn(x))+ gn(x) ,
304 Lucio Boccardo, Juan Casado-Diaz and Luigi Orsina
where
Gn(x) = un
1+ 1n |un |
En(x)
1+ 1n |∇un |
, gn(x) = fn(x)− A En(x)
1+ 1n |un |
· ∇un
1+ 1n |∇un |
.
Since
|Gn(x)| ≤ n2 , |gn(x)| ≤ n2 ,
from Lax–Milgram theorem and from a result by Stampacchia (see [12], Théorème 4.1), it follows
that there exists C > 0 such that
∥un∥
W 1,20 (Ω)
+∥un∥
L∞(Ω)
≤C n2.
Since every un is a bounded function, it is possible to use a nonlinear composition of un as test
function in (14).
In the following, we recall that the Sobolev exponent 2∗ is 2N /(N −2), if N > 2. For N = 2, we
define 2∗ as a positive number bigger than 2 to be chosen later.
Step 1. In this step, we assume A ≥ 1 and we will prove the existence of finite energy solutions.
We use |un |A−1un as test function in (14), and we have
A
∫
Ω
M(x)∇un ·∇un |un |A−1 − A
∫
Ω
un
1+ 1n |un |
En(x)
1+ 1n |∇un |
·∇un |un |A−1
+ A
∫
Ω
un
1+ 1n |un |
En(x)
1+ 1n |∇un |
·∇un |un |A−1 =
∫
Ω
fn(x) |un |A−1un . (15)
Thus, after cancellation of equal terms, we have that
A
∫
Ω
M(x)∇un ·∇un |un |A−1 =
∫
Ω
fn(x) |un |A−1un ,
which, by (4) and Sobolev’s inequality, implies that
4Aα
(A+1)2
[∫
Ω
|un |
2∗
2 (A+1)
] 2
2∗ ≤S Aα
∫
Ω
|∇un |2|un |A−1 ≤S ∥ f ∥
Lm (Ω)
[∫
Ω
|un |A m
′
] 1
m′
. (16)
If N > 3, the condition 2∗2 (A + 1) = A m′ holds if m = N (A+1)N+2A (which is our assumption); thus,
from (16) we obtain that
4Aα
(A+1)2 ∥un∥L 2∗2 (A+1)(Ω) ≤S ∥ f ∥Lm (Ω). (17)
If N = 2, taking into account that 2∗ can be chosen arbitrarily large, we deduce that m can be
chosen any number bigger than one in order to have again (17).
Thus, we proved that
the sequence {un} is bounded in L
2∗
2 (A+1)(Ω). (18)
Using this result in (16) yields that
the sequence {|un |
A−1
2 un} is bounded in W
1,2
0 (Ω). (19)
We now use T1(un) as test function in (14) and we have (thanks to Young’s inequality)
α
∫
Ω
|∇T1(un)|2
≤
∫
Ω
|un ||E(x)||∇T1(un)|+ A
∫
Ω
|E(x)||∇un |+
∫
Ω
| f (x)|
≤ (A+1)
∫
Ω
|E(x)||∇T1(un)|+ A
∫
Ω
|E(x)||∇T1(un)|+ 2A
A+1
∫
Ω
|E(x)||∇(|un |
A−1
2 un)|+
∫
Ω
| f (x)|
≤ α
2
∫
Ω
|∇T1(un)|2 +
∫
Ω
| f (x)|+C
(∫
Ω
|E(x)|2 +
∫
Ω
∣∣∇(|un | A−12 un)∣∣2)
Lucio Boccardo, Juan Casado-Diaz and Luigi Orsina 305
which proves that the sequence {T1(un)} is bounded in W
1,2
0 (Ω). Combined with (19), we get that
the sequence {un} is bounded in W
1,2
0 (Ω). (20)
The reflexivity of W 1,20 (Ω) and Rellich theorem then imply the existence of a subsequence {un j }
and a function u in W 1,20 (Ω), with |u|
A−1
2 u in W 1,20 (Ω), such that
un j * u in W
1,2
0 (Ω)
|un j |
A−1
2 un j * |u|
A−1
2 u in W 1,20 (Ω),
un j → u in Lρ(Ω), 1 ≤ ρ <
2∗
2
(A+1).
(21)
Note that the boundedness of {un} in W
1,2
0 (Ω) implies that ∥ 1n ∇un∥(L2(Ω))N → 0 which implies
(up to a subsequence) that 1n ∇un(x) → 0 a.e. in Ω. Thus we can pass to the limit in (14) to prove
that u is a weak solution of (2); that is u belongs to W 1,20 (Ω) and is such that∫
Ω
M(x)∇u ·∇v −
∫
Ω
u [E(x) ·∇v]+ A
∫
Ω
[E(x) ·∇u] v =
∫
Ω
f v, ∀ v ∈C 10 (Ω). (22)
Step 2. In this step, we assume 0 < A < 1 and we will prove the existence of infinite energy
solutions.
Since 0 < A < 1, we need to modify our test function, and use [(h +|un |)A −h A]sign(un), with
h > 0. Then we have
A
∫
Ω
M(x)∇un ·∇un (h +|un |)A−1 − A
∫
Ω
un
1+ 1n |un |
En(x)
1+ 1n |∇un |
·∇un(h +|un |)A−1
+ A
∫
Ω
1
1+ 1n |un |
En(x)
1+ 1n |∇un |
·∇un [(h +|un |)A −h A]sign(un) ≤
∫
Ω
| f |(h +|un |)A .
Since
(h +|un |)Asign(un) = (h +|un |)A−1
(
h sign(un)+un
)
,
the above identity implies that
A
∫
Ω
M(x)∇un ·∇un (h +|un |)A−1 + A
∫
Ω
1
1+ 1n |un |
En
1+ 1n |∇un |
·∇un [(h +|un |)A−1h −h A]sign(un)
≤
∫
Ω
| f (x)|(h +|un |)A .
Since 0 < A < 1, one has that
(h +|un |)A−1h ≤ h A ,
so that we can use Lebesgue theorem to pass to the limit as h tends to zero in the second term.
Using also the monotone convergence theorem in the first one and the Lebesgue theorem in the
third one, we get
A
∫
Ω
M(x)∇un ·∇un
|un |1−A
≤
∫
Ω
| f (x)||un |A . (23)
Thus, the inequalities in (16) still hold. As above, this proves (18) and (19). For 1 ≤ r ′ < 2, Hölder’s
inequality also gives∫
Ω
|∇un |r
′ =
∫
Ω
|∇un |r ′
|un |
r ′(1−A)
2
|un |
r ′(1−A)
2 ≤
(∫
Ω
|∇un |2
|un |1−A
) r ′
2
(∫
Ω
|un |
r ′(1−A)
2−r ′
)2− r ′2
.
306 Lucio Boccardo, Juan Casado-Diaz and Luigi Orsina
Since r
′(1−A)
2−r ′ = 2
∗
2 (1+A), in the case N ≥ 3, from (18) and (19), it follows that the sequence {un}
is bounded in W 1,r
′
0 (Ω). The same result is true in the case N = 2 if we define 2∗ > 2 (recall that
1+ A < r ′ < 2 by (5)) by
2∗ = 2 r
′(1− A)
(2− r ′)(1+ A) .
Thus it is possible to pass to the limit as in the first step to get that u is a solution of (2). □
We now prove that, under the assumption that f (x) ≥ 0, the solution u is not only positive, but
cannot be zero in a set of positive measure.
Proposition 4. If f (x) ≥ 0, then u(x) ≥ 0. Moreover, if f (x) is not identically zero, then u(x) can be
zero at most in a set of zero measure.
Proof. We give the proof in the case A ≥ 1; the case 0 < A < 1 can be proved modifying the test
function, as in Step 2 of the proof of Theorem 1. Choosing v = |un |A−1u−n as test function in (14)
we obtain identity (15) with u−n instead of un . Thus, one can cancel two equal terms to obtain
that
−A
∫
Ω
M(x)|un |A−1∇u−n ·∇u−n = A
∫
Ω
M(x)|un |A−1∇un ·∇u−n =
∫
Ω
fn(x)u
−
n |un |A−1.
Since the right hand side is positive, and the left hand side is negative, one has that u−n = 0, so that
un ≥ 0. Recalling that u is the limit of the sequence {un}, we have proved that u(x) ≥ 0.
The second statement can be proved exactly as in [5, Theorem 3.1 and Theorem 4.1]. □
References
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(1989), no. 1, p. 149-169.
[8] M. Briane, J. Casado-Díaz, “A class of second-order linear elliptic equations with drift: renormalized solutions,
uniqueness and homogenization”, Potential Anal. 43 (2015), no. 3, p. 399-413.
[9] M. Briane, P. Gérard, “A drift homogenization problem revisited”, Ann. Sc. Norm. Super. Pisa, Cl. Sci. 11 (2012), no. 1,
p. 1-39.
[10] G. Dal Maso, F. Murat, L. Orsina, A. Prignet, “Renormalized solutions of elliptic equations with general measure data”,
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[11] D. Gilbarg, N. S. Trudinger, Elliptic partial differential equations of second order, Classics in Mathematics, Springer,
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Dirichlet problems with skew-symmetric drift terms

Niels Lubbes

Translational and great Darboux cyclides

Shi-Qiang Chen

Yuchen Ding

Xiaodong Lü

Yuhan Zhang

Comptes Rendus
Mathématique
Shi-Qiang Chen, Yuchen Ding, Xiaodong Lü and Yuhan Zhang
Correct order on some certain weighted representation functions
Volume 362 (2024), p. 547-552
https://doi.org/10.5802/crmath.573
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
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Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 547-552
https://doi.org/10.5802/crmath.573
Research article / Article de recherche
Combinatorics, number theory / Combinatoire, théorie des nombres
Correct order on some certain weighted
representation functions
L’ordre correct de certaines fonctions de représentation
pondérées
Shi-Qiang Chen a, Yuchen Ding ∗,b, Xiaodong Lü b and Yuhan Zhang b
a School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002, People’s Republic of China
b School of Mathematical Sciences, Yangzhou University, Yangzhou 225002, People’s Republic of China
E-mails: csq20180327@163.com (S.-Q. Chen), ycding@yzu.edu.cn (Y. Ding), xdlv@yzu.edu.cn (X. Lü),
Qiaoyuan0804@hotmail.com (Y. Zhang)
Abstract. Let N be the set of all nonnegative integers. For any positive integer k and any subset A of
nonnegative integers, let r1,k (A,n) be the number of solutions (a1, a2) to the equation n = a1 +ka2. In 2016,
Qu proved that
liminf
n→∞ r1,k (A,n) =∞
providing that r1,k (A,n) = r1,k (N\ A,n) for all sufficiently large integers, which answered affirmatively a 2012
problem of Yang and Chen. In a very recent article, another Chen (the first named author) slightly improved
Qu’s result and obtained that
liminf
n→∞
r1,k (A,n)
logn
> 0.
In this note, we further improve the lower bound on r1,k (A,n) by showing that
liminf
n→∞
r1,k (A,n)
n
> 0.
Our bound reflects the correct order of magnitude of the representation function r1,k (A,n) under the above
restrictions due to the trivial fact that r1,k (A,n) ≤ n/k.
Résumé. SoitN l’ensemble de tous les entiers non négatifs. Pour tout entier positif k et tout sous-ensemble A
d’entiers non négatifs, notons r1,k (A,n) le nombre de solutions (a1, a2) de l’équation n = a1 +ka2. En 2016,
Qu a prouvé que
liminf
n→∞ r1,k (A,n) =∞
ce qui signifie que r1,k (A,n) = r1,k (N \ A,n) pour tous les entiers suffisamment grands, ce qui répondait par
l’affirmative à un problème de Yang et Chen datant de 2012. Dans un article très récent, un autre Chen (le
premier auteur dans notre article) a légèrement amélioré le résultat de Qu et obtenu que
liminf
n→∞
r1,k (A,n)
logn
> 0.
Dans cette note, nous améliorons encore le minorant de r1,k (A,n) en montrant que
liminf
n→∞
r1,k (A,n)
n
> 0.
Notre limite reflète l’ordre de grandeur correct de la fonction de représentation r1,k (A,n) sous les restrictions
ci-dessus en raison du fait trivial que r1,k (A,n) ≤ n/k.
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
548 Shi-Qiang Chen, Yuchen Ding, Xiaodong Lü and Yuhan Zhang
Keywords. representation functions, order of functions, partitions of integers.
Mots-clés. fonctions de représentation, ordre des fonctions, partitions d’entiers.
2020 Mathematics Subject Classification. 11B34, 11A41.
Funding. The first author is supported by the National Natural Science Foundation of China (Grant No.
12301003), the Anhui Provincial Natural Science Foundation (Grant No. 2308085QA02) and the University
Natural Science Research Project of Anhui Province (Grant No. 2022AH050171). The second author is
supported by National Natural Science Foundation of China under Grant No. 12201544, Natural Science
Foundation of Jiangsu Province, China, Grant No. BK20210784, China Postdoctoral Science Foundation,
Grant No. 2022M710121, the foundations of the projects “Jiangsu Provincial Double–Innovation Doctor
Program”, Grant No. JSSCBS20211023 and “Golden Phoenix of the Green City–Yang Zhou” to excellent
PhD, Grant No. YZLYJF2020PHD051. The fourth author is supported by Yangzhou University Science and
Technology Innovation foundation Program for College Students (Grant No. XCX20230276), University Brand
Major Construction Foundation Program of Jiangsu Province (Mathematics and Applied Mathematics, Grant
No. PPZY2015B109).
The third author is supported by National Natural Science Foundation of China under Grant No. 12101538.
Manuscript received 29 June 2023, revised 17 August 2023, accepted 19 September 2023.
1. Introduction
Let N be the set of all nonnegative integers and A a subset of N. For any nonnegative integer n,
let R1(A,n);R2(A,n) and R3(A,n) be the number of solutions (a, a′) to the equations n = a + a′
with a, a′ ∈ A; a, a′ ∈ A, a < a′ and a, a′ ∈ A, a ≤ a′, respectively. For backgrounds on these
representation functions Ri (A,n), i = 1,2,3, one can refer to an early survey article of Sárközy
and Sós [11]. Following Sárközy’s question, Dombi [6], Chen and Wang [5], Lev [7], Sándor [10],
Tang [13], Chen and Tang [4] and Chen [3] investigated various properties on values of the
representation functions Ri (A,n) and Ri (N\ A,n), i = 1,2,3.
In an interesting paper, Yang and Chen [15] introduced the following weighted representation
function
rk1,k2 (A,n) = #
{
(a1, a2) ∈ A2 : n = k1a1 +k2a2
}
,
where A is a subset of N and k1,k2 are two positive integers. They determined all pairs (k1,k2) of
positive integers for which there exists a set A ⊆N such that
rk1,k2 (A,n) = rk1,k2 (N\ A,n)
for all sufficiently large integers, which would reduce to partial answers to the original question
of Sárközy mentioned above for k1 = k2 = 1 on R1(A,n). For 1 ≤ k1 < k2 with (k1,k2) = 1, if there
exists a set A ⊆N such that
rk1,k2 (A,n) = rk1,k2 (N\ A,n)
for all sufficiently large integers, then Yang and Chen proved that k1 = 1. So the studies of the
weighted representation function r1,k (A,n) would be of particular interest.
For a positive integer k ≥ 2, letΨk be the set of all A ⊆N such that
r1,k (A,n) = r1,k (N\ A,n)
for sufficiently large integers n. A result of Yang [14] states that if k,ℓ are multiplicatively
independent (equivalently, logk/logℓ is irrational), then Ψk ∩Ψℓ = ;. Qu [9] then gave a
complete criteria for which Ψk ∩Ψℓ = ;. It turns out to be that Ψk ∩Ψℓ ̸= ; if and only
if logk/logℓ = a/b for some odd positive integers a and b, which disproved a conjecture of
Yang [14]. For related result, see also the article of Li and Ma [8], Shallit [1, 12]. In [15], Yang
and Chen posed the following problem:
Shi-Qiang Chen, Yuchen Ding, Xiaodong Lü and Yuhan Zhang 549
Problem 1. For any set A ∈Ψk , is it true that r1,k (A,n) ≥ 1 for all sufficiently large integers n? Is it
true that r1,k (A,n) →∞ as n →∞?
Problem 1 was later answered affirmatively by Qu [9]. Very recently, Chen [2] improved Qu’s
result by showing that
liminf
n→∞
r1,k (A,n)
logn
> 0
for any A ∈Ψk . In this note, we give the following very much stronger bound.
Theorem 2. Let k ≥ 2 be a given integer. For any set A ∈Ψk , we have
liminf
n→∞
r1,k (A,n)
n
> 0.
Remark 3. It can be seen that our new bound is sharp on the order of magnitude of r1,k (A,n) in
the sense that
limsup
n→∞
r1,k (A,n)
n
≤ 1/k.
Perhaps, it should be pointed out that the original argument of Qu [9] with necessary adjustments
can also lead to the bound given by Chen [2]. It is also worth mentioning that our argument here
leading to the sharp bound above in Theorem 2 is different and simplified comparing with the
somewhat complicated ones taken by Qu and Chen.
2. Proofs
Following Qu [9], we may write A as the following union of the “blocks”
A =
∞⋃
i=0
[t2i , t2i+1) ,
where 0 ≤ t0 < t1 < t2 < · · · is an increasing sequence of integers. The proof of our theorem is
based on the following lemma of Qu [9, Lemma 2.1].
Lemma 4. Let k ≥ 2 be a given integer. For any A ∈Ψk with
A =
∞⋃
i=0
[t2i , t2i+1) ,
there exist an odd positive integer a and a nonnegative integer i0 such that ti+a = kti for all i ≥ i0.
Proof of Theorem 2. By Lemma 4 for A ∈Ψk with
A =
∞⋃
i=0
[t2i , t2i+1) ,
there exists an odd positive integer a such that ti+a = kti for all i ≥ i0. Without loss of generality,
we can assume that i0 = 0, otherwise one can consider Ã = A \ [0, ti0 ) instead of A (this can be
seen from the proofs below). Let T = 4(ta+2 − t0). Then there exists some odd integer g ∈N such
that kg > T .
From now on, let n be a sufficiently large number. It is clear that there are nonnegative integers
m and r with 0 ≤ r < (kg +1) such that
n = (kg +1)m + r.
We can assume that m ∈ [k s tℓ,k s tℓ+1) for two nonnegative integers s and ℓ with
0 ≤ ℓ≤ a −1.
550 Shi-Qiang Chen, Yuchen Ding, Xiaodong Lü and Yuhan Zhang
Recall that the integer n is assumed to be sufficiently large, it follows that both m and s are
sufficiently large. We will prove that
r1,k (A,n) ≥
n
k5ta (kg +2)
− (kg +1) ,
from which it follows clearly that
liminf
n→∞
r1,k (A,n)
n
> 0.
Since r1,k (A,n) = r1,k (N \ A,n) for all sufficiently large n, we can also assume, without loss of
generality, that [
k s tℓ,k
s tℓ+1
)⊆ A.
Since g is an odd integer, we make the observation that each of the following three intervals[
k s tℓ−2,k s tℓ−1
)
,
[
k s tℓ+2,k s tℓ+3
)
and
[
k s+g−1tℓ,k s+g−1tℓ+1
)
contains in A as well, which is crucial in the following arguments. Before the continuation of the
proof, we make the following notice that for brevity we write, for example,
k3t2, k
4t2−a and k2t2+a
as the same number at different occasions. The proofs are divided into three cases:
Case I. k s tℓ+k s−4 ≤ m < k s tℓ+1 −k s−4. Noting that for any 0 ≤ q < k s−5 − r , we have
n = (m +kq + r )+k (kg−1m −q) ,
where both m +kq + r and kg−1m −q belong to A since
m +kq + r ∈ [k s tℓ,k s tℓ+1) and kg−1m −q ∈ [k s+g−1tℓ,k s+g−1tℓ+1) .
Thus, we deduce that
r1,k (A,n) ≥ k s−5 − r
> m
k5tℓ+1
− (kg +1)
≥ n − r
k5ta (kg +1)
− (kg +1)
≥ n
k5ta (kg +2)
− (kg +1) .
Case II. k s tℓ ≤ m < k s tℓ+k s−4. For any
k s−1 (tℓ− tℓ−1)+k s−5 + r < q ≤ k s−1 (tℓ− tℓ−2) ,
it can be seen that
m −kq + r ∈ [k s tℓ−2,k s tℓ−1)⊆ A
and
kg−1m +q ∈ [k s+g−1tℓ,k s+g−1tℓ+1)⊆ A,
where the latter inclusion relation comes from the observation that [k s+g−1tℓ,k s+g−1tℓ+1) con-
tains in A made previously and the facts that
kg−1m +q < kg−1 (k s tℓ+k s−4)+k s−1 (tℓ− tℓ−2) ≤ k s+g−1tℓ+1
since kg > T ≥ 2(tℓ− tℓ−2). Note that
n = (m −kq + r )+k (kg−1m +q)
for all these q , from which we conclude that
r1,k (A,n) ≥ k s−1(tℓ− tℓ−2)−k s−1(tℓ− tℓ−1)−k s−5 − r
≥ 1
2
k s−1 − r
Shi-Qiang Chen, Yuchen Ding, Xiaodong Lü and Yuhan Zhang 551
≥ n − r
2kta (kg +1)
− (kg +1)
≥ n
2kta (kg +2)
− (kg +1) .
Case III. k s tℓ+1 −k s−4 ≤ m < k s tℓ+1. It can be verified directly that
m +kq + r ∈ [k s tℓ+2,k s tℓ+3)⊆ A
and
kg−1m −q ∈ [k s+g−1tℓ,k s+g−1tℓ+1)⊆ A,
for any
k s−1 (tℓ+2 − tℓ+1)+k s−5 ≤ q ≤ k s−1 (tℓ+3 − tℓ+1)− r
via similar arguments in Case II. In fact,
kg−1m −q ≥ kg−1 (k s tℓ+1 −k s−4)−k s−1 (tℓ+3 − tℓ+1) ≥ k s+g−1tℓ
since kg > T ≥ 2(tℓ+3 − tℓ+1). Note that
n = (m +kq + r )+k (kg−1m −q)
for all these q , from which we conclude that
r1,k (A,n) ≥ k s−1(tℓ+3 − tℓ+1)−k s−1 (tℓ+2 − tℓ+1)−k s−5 − r
≥ 1
2
k s−1 − r
≥ n − r
2kta (kg +1)
− (kg +1)
≥ n
2kta (kg +2)
− (kg +1) .
This completes the proof of Theorem 2. □
Acknowledgments
The authors would like to thank the anonymous referee for his helpful comments.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
References
[1] J. P. Bell and J. Shallit, “Counterexamples to a conjecture of Dombi in additive number
theory”, Acta Math. Hung. 169 (2023), pp. 562–565.
[2] S.-Q. Chen, “The lower bound of weighted representation function”, 2306.16025.
(preprint), to appear in Period. Math. Hungar.
[3] Y.-G. Chen, “On the values of representation functions”, Sci. China, Math. 54 (2011), no. 7,
pp. 1317–1331.
[4] Y.-G. Chen and M. Tang, “Partitions of natural numbers with the same representation
functions”, J. Number Theory 129 (2009), pp. 2689–2695.
[5] Y.-G. Chen and B. Wang, “On additive properties of two special sequences”, Acta Arith. 110
(2003), pp. 299–303.
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[6] G. Dombi, “Additive properties of certain sets”, Acta Arith. 103 (2002), pp. 137–146.
[7] V. F. Lev, “Reconstructing integer sets from their representation functions”, Electron. J.
Comb. 11 (2004), no. 1, article no. R78.
[8] Y.-L. Li and W.-X. Ma, “Partitions of natural numbers with the same weighted representa-
tion functions”, Colloq. Math. 159 (2020), pp. 1–5.
[9] Z. Qu, “A note on representation functions with different weights”, Colloq. Math. 143
(2016), pp. 105–112.
[10] C. Sándor, “Partitions of natural numbers and their representation functions”, Integers 4
(2004), article no. A18.
[11] A. Sárközy and V. T. Sós, “On additive representation functions”, in The mathematics of Paul
Erdős. Vol. I. (R. L. Graham et al., ed.), Springer, 1997, pp. 129–150.
[12] J. Shallit, “A Dombi counterexample with positive lower density”, Integers 23 (2023), article
no. #A74.
[13] M. Tang, “Partitions of the set of natural numbers and their representation functions”,
Discrete Math. 308 (2008), pp. 2614–2616.
[14] Q.-H. Yang, “Representation functions with different weights”, Colloq. Math. 137 (2014),
pp. 1–6.
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sentation functions”, J. Number Theory 132 (2012), pp. 3047–3055.


Correct order on some certain weighted representation functions

Jason Bell

Chris Schulz

Jeffrey Shallit

Comptes Rendus
Mathématique
Jason Bell, Chris Schulz and Jeffrey Shallit
Consecutive Power Occurrences in Sturmian Words
Volume 362 (2024), p. 1273-1278
Online since: 5 November 2024
https://doi.org/10.5802/crmath.644
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
C EN T R E
MER S ENN E
The Comptes Rendus. Mathématique are a member of the
Mersenne Center for open scientific publishing
www.centre-mersenne.org — e-ISSN : 1778-3569
Comptes Rendus. Mathématique
2024, Vol. 362, p. 1273-1278
https://doi.org/10.5802/crmath.644
Research article / Article de recherche
Theoretical computer science / Informatique théorique
Consecutive Power Occurrences in Sturmian
Words
Occurrences consécutives de puissance dans les mots
sturmiens
Jason Bell a, Chris Schulz a and Jeffrey Shallit ∗,b
a University of Waterloo, Department of Pure Mathematics, 200 University Avenue West,
Waterloo, Ontario N2L 3G1, Canada
E-mails: jpbell@uwaterloo.ca, chris.schulz@uwaterloo.ca, shallit@uwaterloo.ca
Abstract. We show that every Sturmian word has the property that the distance between consecutive ending
positions of cubes occurring in the word is always bounded by 10 and this bound is optimal, extending a
result of Rampersad, who proved that the bound 9 holds for the Fibonacci word. We then give a general
result showing that for every e ∈ [1, (5+p5)/2) there is a natural number N , depending only on e, such that
every Sturmian word has the property that the distance between consecutive ending positions of e-powers
occurring in the word is uniformly bounded by N .
Résumé. Nous montrons que la distance entre deux positions finales consécutives de cubes apparaissant
dans un mot sturmien est toujours inférieure ou égale à 10 et que cette valeur est optimale, étendant ainsi un
résultat de Rampersad, qui a démontré que cette distance est majorée par 9 pour le mot de Fibonacci. Nous
donnons ensuite un résultat général montrant que pour tout e ∈ [1, (5+p5)/2) il existe un entier naturel N ,
dépendant uniquement de e, tel que la distance entre deux positions finales consécutives de puissances e
apparaissant dans un mot sturmien est uniformément majorée par N .
Keywords. Sturmian word, cube, periodicity, balanced word.
Mots-clés. Mot Sturmien, cube, périodicité, mot équilibré.
2020 Mathematics Subject Classification. 68R15.
Funding. The authors were supported by Discovery Grants from the National Sciences and Engineering
Research Council of Canada.
Manuscript received 19 February 2024, revised 26 March 2024, accepted 21 April 2024.
1. Introduction
In this note we are concerned with the occurrences of powers in words. We say that a length-n
word w = w[0..n − 1] has period p if w[i ] = w[i + p] for i ∈ {0, . . . ,n − p − 1}. The smallest such
period is called the period and is written per(w). The exponent of a length-n word w is defined
to be exp(w) = n/per(w). For example, exp(entente) = 7/3. For a real number e ≥ 1, we say a
word w is an e-power if ⌈e per(w)⌉ is equal to |w |, the length of w . In particular, a finite word
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1274 Jason Bell, Chris Schulz and Jeffrey Shallit
being an e-power and having exponent e are not in general the same: the exponent is necessarily
rational, while there exist e-powers for every real number e ≥ 1. A 2-power is called a square, like
the English word murmur. A 3-power is called a cube, like the English word shshsh.
Letα= (1+p5)/2 be the golden ratio and let x be any infinite word. In 1997 the third author of
this paper conjectured, and Mignosi, Restivo, and Salemi later proved [10], that every sufficiently
long prefix of x has a suffix of exponent ≥ α2 if and only if x is ultimately periodic. This is an
example of the “local periodicity implies global periodicity” phenomenon.
Furthermore, the constant α2 is best possible. Let f = f[0]f[1]f[2] · · · = 01001010 · · · denote the
infinite Fibonacci word [4], the fixed point of the morphism sending 0 → 01 and 1 → 0. Mignosi,
Restivo, and Salemi [10] also showed that for all ϵ> 0 every sufficiently long prefix of f has a suffix
that is anα2−ϵ power. For example, there is a square suffix of every prefix of length n ≥ 6 in f, and
there is a 52 -power suffix of every prefix of length n ≥ 220 in f. An explicit version of this result was
recently proved by the third author [13].
More generally, given an infinite word x and an exponent e, one can consider the list of all
positive integers p1 < p2 < ·· · marking the position where some e-power in x ends. For example,
for the Fibonacci word we have
• (010)3 ends at position 13
• (01001)3 ends at position 22
• (10010)3 ends at position 23
• (010)3 ends at position 26
• (010)3 ends at position 34
and so forth.
Recently Rampersad [12] obtained an explicit description of the positions where cubes end in
the Fibonacci word f.
In particular, he observed that if p1 < p2 are two consecutive positions where cubes end in f,
then p2 −p1 ≤ 9, and more precisely p2 −p1 ∈ {1,2,3,4,8,9}. (Note that he phrased his discussion
in terms of the lengths of runs of consecutive positions where there are no cubes ending, so the
numbers he presented differ by 1 from ours.)
Since the Fibonacci word is the simplest of a much larger class of binary infinite words called
Sturmian words, this naturally raises the question of whether a similar result holds for this much
larger class. Sturmian words have several equivalent descriptions, as follows:
(a) Infinite words of the form (⌊(n+1)γ+β⌋−⌊nγ+β⌋)n≥1 for real 0 < γ< 1, 0 ≤β< 1, where
γ is irrational. Here γ is called the slope of the word and β is called the intercept.
(b) Infinite words having exactly n+1 distinct factors of length n for all n ≥ 0. Here by a factor
we mean a consecutive block lying inside the word.
(c) Infinite aperiodic binary balanced words, that is, words such that for all factors x, y of the
same length, and all a ∈ {0,1}, the inequality ∣∣|x|a −|y |a∣∣≤ 1 holds, where |z|a denotes the
number of occurrences of the letter a in z.
For more information about this class of words, see, for example, [9, Chapter 2].
In this note we generalize Rampersad’s result to all Sturmian words.
Theorem 1. Let x be a Sturmian word. Then the maximum gap between positions where cubes
end in x is at most 10, and this bound is optimal.
In fact, our proof shows that the gap between consecutive occurrences of cubes having period
at most 5 is at most 10 for every Sturmian word; furthermore, the proof shows that the optimal
bound 10 is achieved by the Sturmian characteristic word with slope
p
2−1 and intercept 0.
It is natural to ask whether a similar result to Theorem 1 holds for powers other than cubes. It
is known that the Fibonacci word does not haveβ-powers forβ= (5+p5)/2, and so it is necessary
Jason Bell, Chris Schulz and Jeffrey Shallit 1275
to restrict to exponents that are less than β when considering the gap question over all Sturmian
words. Once this condition is imposed, however, we are able to again prove a general result about
the existence of uniform bounds on gaps of e-powers in Sturmian words for every e <β.
Theorem 2. Let x be a Sturmian word and let e ∈ [1, (5+p5)/2). Then there are natural numbers N
and k, depending only on e, such that the maximum gap between the ending positions of e-powers
of period at most k in x is at most N .
2. Proof of Theorem 1
We begin with a lemma. As it turns out, it suffices to consider words of bounded period.
Lemma 3. Every balanced binary word of length 17 contains a cube of period at most 5, and the
bound 17 is best possible.
Proof. We enumerate all 594 binary balanced words of length 17 and check. For length 16, the
word 0010100101001001 is balanced but contains no cube. □
Proof of Theorem 1. We claim that every balanced binary word of length 32 contains at least two
consecutive cube occurrences, each of period at most 5. From Lemma 3, we know the prefix of
length 17 contains a cube of period at most 5, and so does the suffix of length 17. Since in a
word of length 32 these two cubes can overlap in at most two symbols, they must be distinct cube
occurrences. Thus the maximum possible gap must appear in some balanced word of length 32.
It then suffices to examine all balanced words of this length (there are 3650) and compute all gaps
between consecutive ending positions of cubes of period 5 in all of them. The longest is 10.
To prove that 10 is optimal, we use the theorem-prover Walnut; see [11, 14] for more details
about it. We now consider the Sturmian characteristic word with slope
p
2 − 1 and intercept
0. This word has an associated Ostrowski numeration system based on the Pell numbers,
1,2,5,12,29,70,169, . . .; this system is built-in to Walnut and is invoked by saying msd_pell (see
[1–3]). We now use the following Walnut code:
reg odd0 msd_pell "(0|1|2)*(1|2)0(00)*":
def root2 "?msd_pell (n>=0) & $odd0(n+1)":
combine RT2 root2:
# make a DFAO RT2 for Sturmian sequence w/slope sqrt(2)-1
def end_in_cube "?msd_pell Ei,j (j>=1) & (i+3*j=n) &
At (t<2*j) => RT2[i+t]=RT2[i+j+t]":
# is there a cube ending at position n
def two_consec "?msd_pell (i<j) & $end_in_cube(i) &
$end_in_cube(j) & At (i<t & t<j) => ~$end_in_cube(t)":
# are i and j two consecutive cube-ending positions?
def two_consec_dist "?msd_pell (n>0) & Ei,j i+n=j & $two_consec(i,j)":
# it accepts only 1, 110 and 200, i.e., 1, 7 and 10 in Pell
The code first creates a deterministic finite-state automaton with output, named RT2, for the
Sturmian sequence x with slope
p
2 − 1 and intercept 0. It next determines the set of natural
numbers n for which there is a cube occurring in x that ends at position n. Finally, the code finds
the set of gaps between consecutive ending positions of cubes and determines that the only gaps
that occur are the natural numbers whose expansions in the Pell numeration system are 1,110,
and 200. These are the numbers 1,7,10 and so the bound of 10 is best possible. □
1276 Jason Bell, Chris Schulz and Jeffrey Shallit
Remark 4. We note that the proof in fact shows the slightly stronger claim that the gap between
consecutive occurrences of cubes of period at most 5 is at most 10 for every Sturmian word
and that the bound 10 is achieved by the Sturmian characteristic word with slope
p
2− 1 and
intercept 0.
3. General exponents
We now give the proof of Theorem 2. We recall that a partial function f from a set X to a set Y is
simply a map defined on a non-empty subset U of X that maps U into the set Y . In the case that
the map is defined on all of X , the map is called a total function. We write f : X * Y for a partial
function from X to Y .
Given two partial functions f , g : N* R≥0, we declare that f ⪯ g if f (i ) ≤ g (i ) for all i in the
domain of both f and g . We note that ⪯ is not a partial order on partial functions, but it is,
however, a partial order on total functions. We recall that for a right-infinite word x over a finite
alphabet, we have a subword complexity function px whose value at n is the number of distinct
length-n factors of x. If w is a finite word, we have a partial function pw : N* N that is defined
on the nonnegative integers that are less than or equal to the length of w and whose value at i is
the number of factors of w of length i for all i less than or equal to the length of w .
Recall that an infinite word x is uniformly recurrent if for each of its finite factors v , there is an
integer n(v) such that every block of length n(v) in x contains a copy of v .
Lemma 5. Let Σ be a finite alphabet, let h : N→ N be a total function, and let S be a non-empty
set of words over Σ. Suppose that every uniformly recurrent word x with px ⪯ h has the property
that x contains some word from S as a factor. Then there exists a finite subset S0 of S and a natural
number N , depending only on h and S0, such that, whenever x is a right-infinite word with px ⪯ h,
every factor of x of length N contains a factor from S0.
Proof. Let Un denote the set of words u over Σ of length n such that pu ⪯ h and such that u does
not have an element of S as a factor. Then
⋃
n≥0 Un is closed under the process of taking factors
(where we take U0 to be the empty word). If there is some N ≥ 1 such that UN is empty, then
letting S0 be the set of words in S of length at most N , we see that whenever x is a right-infinite
word with px ⪯ h, every factor of x of length N contains a factor from S0.
Thus we may assume that each Un is non-empty. By König’s infinity lemma (see [8] or [7,
Section 2.3.4.3]), there is a right-infinite word x with the property that all of its finite factors are
in the union of the Ui . In particular, px ⪯ h and x has no words from S as a factor. Then by
Furstenberg’s theorem [6], there is a uniformly recurrent right-infinite word y whose finite factors
are all factors of x, so py ⪯ px ⪯ h and y has no occurrences of factors from S, a contradiction. □
Proof of Theorem 2. Let h(n) = n +1 for n ≥ 0 and let Pe denote the set of words w over Σ that
are e-powers. By work of Damanik and Lenz [5], every Sturmian word has an element from Pe
as a factor. Since every uniformly recurrent word whose subword complexity function is ⪯ h is
either Sturmian or periodic and since periodic words always contain elements from Pe as factors,
Lemma 5 gives that there is a finite subset Qe of Pe and some number n = n(e) such that every
length-n factor of a Sturmian word x has an occurrence of a word from Qe . In particular, if we
take k to be the supremum of periods in Qe and N = 2n, we see that gaps between consecutive
e-powers of period at most k are bounded by N . □
4. Other exponents
When one considers Theorem 2, a natural question that arises is whether one can bound the
associated quantities k and N explicitly in terms of the exponent e ∈ [1, (5+p5)/2). This appears
Jason Bell, Chris Schulz and Jeffrey Shallit 1277
to be somewhat subtle in general, but using the same computational approach as we employed
in establishing Theorem 1, we can obtain results for several other exponents.
For each exponent e we report the smallest n such that such that every balanced word of length
n contains a factor that is an e-power, the largest period p of an e-power in those words, the
largest gap g between ending positions of e-powers, and a quadratic irrational γ realizing this
gap g .
Table 1. Data for some exponents
e n p g γ
5/2 9 3 6 (5+p5)/10
8/3 15 5 9
p
2−1
3 17 5 10
p
2−1
16/5 30 8 17 (25−p5)/62
23/7 50 13 27 (59+p5)/158
10/3 69 18 37 (217−p5)/298
It would be interesting to better understand the relationship between the largest gap g and the
exponent e as e approaches (5+p5)/2.
Declaration of interests
The authors do not work for, advise, own shares in, or receive funds from any organization
that could benefit from this article, and have declared no affiliations other than their research
organizations.
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Consecutive Power Occurrences in Sturmian Words

Jean B. Lasserre

Gaussian mixtures closest to a given measure via optimal transport

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