Comptes Rendus Mathématique

Xuan Zhao

Zhenliang Zhang

Comptes Rendus
Mathématique
Xuan Zhao and Zhenliang Zhang
Some lacunarity properties of partial quotients of real numbers
Volume 362 (2024), p. 1089-1095
Online since: 5 November 2024
https://doi.org/10.5802/crmath.594
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. 1089-1095
https://doi.org/10.5802/crmath.594
Research article / Article de recherche
Number theory / Théorie des nombres
Some lacunarity properties of partial
quotients of real numbers
Quelques propriétés de lacunarité des quotients partiels
de nombres réels
Xuan Zhao a and Zhenliang Zhang ∗,b
a National Education Examinations Authority, 100084 Beijing, China
b School of Mathematical Sciences, Chongqing Normal University, Chongqing,
401331, P. R. China
E-mails: zhaox@mail.neea.edu.cn, zhliang_zhang@163.com
Abstract. We consider lacunarity properties of sequence of partial quotients for real numbers in their contin-
ued fraction expansions. Hausdorff dimension of the sets of points with different lacunarity conditions on
their partial quotients are calculated.
Résumé. Nous considérons des propriétés de lacunarité de la suite des quotients partiels du développement
en fraction continue de nombres réels. Nous calculons la dimension de Hausdorff d’ensembles de points
dont la suite des quotients partiels satisfait à différentes conditions de lacunarité.
Keywords. Hausdorff dimension, Continued fraction expansion.
Mots-clés. Dimension de Hausdorff, développement en fraction continue.
Mathematical subject classification (2010). 11K50, 28A78.
Funding. Zhenliang Zhang is supported by project of the Science and Technology Research Program
of Chongqing Municipal Education Commission (No. KJQN202100528), Natural Science Foundation of
Chongqing (No. CSTB2022NSCQ-MSX1255).
Manuscript received 21 July 2023, revised 26 September 2023 and 13 October 2023, accepted 27 November 2023.
1. Introduction
Continued fraction expansion is induced by the Gauss transformation T : [0,1) → [0,1) given by
T (0) := 0, T (x) = 1
x
mod 1, x ∈ (0,1).
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1090 Xuan Zhao and Zhenliang Zhang
Then every irrational number x ∈ [0,1) can be uniquely expanded into an infinite continued
fraction:
x = 1
a1(x)+
1
a2(x)+
. . .
:= [a1(x), a2(x), . . . , ],
where a1(x) = ⌊1/x⌋ and an(x) = a1
(
T n−1(x)
)
for n ≥ 2 are called the partial quotients of x. The
finite truncation
pn(x)
qn(x)
= [a1(x), . . . , an(x)]
is called the nth convergent of x.
It is well-known that continued fraction expansion plays an important role in Diophantine
approximation and dynamical systems:
• In Diophantine approximation, how well an irrational number can be approximated by
rationals depends on the growth rate of the partial quotients. For example, the classic
Jarník set [12] can be expressed as
K (ψ) =
{
x ∈ [0,1) : an+1(x) ≥ψ(qn(x)), for infinitely many n ∈N
}
.
• In dynamical system, continued fraction is a classic dynamics with infinitely many
branches [17].
As said above, the growth rate of the partial quotient is tightly related to the Diophantine
properties of an irrational number. Many metric results have been achieved in this aspect, such
as the Borel–Bernstein theorem that deals with Lebesgue measure theory on the growth rate of
{an(x)}n≥1. Hausdorff dimension of sets obeying some restrictions on the partial quotients have
been well established in Good [6], Łcuzak [16], Wang & Wu [18], etc.
Very recently, it was found by Kleinbock and Wadleigh [14] that the Dirichlet improvable set
is highly related to the growth rate of the product of two consecutive partial quotients in the
following sense. Let ψ : [t0,∞) → R+ be non-increasing with t0 > 1 is fixed and tψ(t ) < 1 for all
t ≥ t0. The ψ-Dirichlet improvavable set is defined as
D(ψ) =
{
x ∈ [0,1) : min
1≤q<Q
∥qx∥ <ψ(Q), for all Q ≫ 1
}
.
Then by takingΨ(q) = qψ(q)1−qψ(q) , one has
G(Ψ) ⊂ [0,1) \D(ψ) ⊂G
(
Ψ
4
)
where
G(Ψ) =
{
x ∈ [0,1) : an+1(x)an(x) ≥Ψ(qn(x)), for infinitely many n ∈N
}
.
Since then, the metric theory relating the growth of two consecutive partial quotients are
extensively studied in [1–3, 7–10, 15].
In this note, we take another turn by studying the relative growth rate of two consecutive
partial quotients. All the above mentioned works are concerning the size of limsup sets, however,
the sets we will consider below are of liminf nature. So the method used here is different from
the above works (see [10] for a unified way of dealing with the Hausdorff dimension of the above
mentioned works). These liminf sets concerns the points with partial quotients increasing very
fast, so can be uniformly well approximated by their convergents. At first, we give some notations.
Let {sn}n≥1 be a sequence of strictly increasing sequence of integers.
Xuan Zhao and Zhenliang Zhang 1091
• call it a sub-lacunary sequence, if
lim
n→∞ sn+1/sn = 1;
• call it lacunary if ∃c > 1 such that
sn+1/sn ≥ c, for all n ≫ 1;
– log-lacunary if for some t > 0,
sn+1
sn
≥ (logn)t , for all n ≫ 1;
– polynomial-lacunary if for some t > 0,
sn+1
sn
≥ nt , for all n ≫ 1;
– exponential-lacunary if for some b > 1,
sn+1
sn
≥ bn , for all n ≫ 1.
At first, we give an auxiliary set to be compliant with the set of points with strictly increasing
partial quotients.
G =
{
x ∈ [0,1) : {an(x)} is a strictly increasing sequence
}
. (1)
Then we define
SG =
{
x ∈G : {an(x)} is sub-lacunary
}
,
LG =
{
x ∈G : {an(x)} is log-lacunary
}
,
PG =
{
x ∈G : {an(x)} is polynomial-lacunary
}
,
EG =
{
x ∈G : {an(x)} is exponential-lacunary
}
.
In this note, we show that
Theorem 1. By denoting dimH the Hausdorff dimension, we have
dimH SG = dimH LG = dimH PG = dimH EG = 1/2.
2. Preliminaries
Recall that pn(x)/qn(x) is the nth convergent of x. The numerator and denominator of
pn(x)/qn(x) can be determined recursively: for any k ≥ 1
pk (x) = ak (x)pk−1(x)+pk−2(x), qk (x) = ak (x)qk−1(x)+qk−2(x) (2)
with the conventions p0 = 0, q0 = 1, p−1 = 1, q−1 = 0.
For simplicity, we write
pn(x) = pn (a1, . . . , an) = pn , qn(x) = qn (a1, . . . , an) = qn (3)
when the partial quotients a1, . . . , an are clear.
For any positive integers a1, . . . , an , define
In (a1, . . . , an) := {x ∈ [0,1) : a1 (x) = a1, . . . , an (x) = an}
and call it a cylinder of order n. We use In(x) to denote the nth order cylinder containing x.
1092 Xuan Zhao and Zhenliang Zhang
Proposition 2 (Khinchin [13]). For any n ≥ 1 and (a1, . . . , an) ∈Nn , pk , qk are defined recursively
by (2) for 0 ≤ k ≤ n. Then
In (a1, . . . , an) =

[
pn
qn
,
pn +pn−1
qn +qn−1
)
if n is even(
pn +pn−1
qn +qn−1
,
pn
qn
]
if n is odd.
(4)
Therefore, the length of a cylinder of order n is given by
|In (a1, . . . , an)| = 1
qn
(
qn +qn−1
) .
The next lemma relates a ball with the cylinders, basically following from Proposition 2 on the
distribution of cylinders [11].
Proposition 3. Let x ∈ In(a1, . . . , an) with an ≥ 2. Then
B(x, |In(a1, . . . , an)|) ⊂
2⋃
i=−1
In(a1, . . . , an + i ).
Next, we introduce the mass distribution principle which is a classic method in estimating the
Hausdorff dimension of a set from below.
Proposition 4 ([4]). Let E be a Borel set and µ be a measure with µ (E) > 0. Suppose that for any
x ∈ E,
liminf
r→0
logµ(B(x,r ))
logr
≥ s (5)
where B (x,r ) denotes an open ball centered at x and radius r , then dimH E ≥ s.
3. Proof of Theorem 1
Good [6] showed that the Hausdorff dimension of G defined in (1) is one-half, so by the simple
inclusion
G ⊃SG , and G ⊃LG ⊃PG ⊃ EG ,
it is sufficient to show that
dimH SG ≥ 1/2 and dimH EG ≥ 1/2.
Lemma 5. Let α> 1. Define
Eα =
{
x ∈ [0,1) : (2n −1)α ≤ an(x) < (2n)α, for all n ≥ 1
}
.
Then
dimH Eα = α−1
2α
.
Proof.
(I) For the upper bound of dimH Eα, we consider a natural cover of Eα. It is clear that for any
N ≥ 1, the family of intervals ⋃
a1,...,aN :(2n−1)α≤an (x)<(2n)α, 1≤n≤N
IN (a1, . . . , aN )
Xuan Zhao and Zhenliang Zhang 1093
covers Eα. Therefore, for any s > 0, the s-Hausdorffmeasure of Eα can be estimated from
above by
H s (Eα) ≤ liminf
N→∞
∑
a1,...,aN :(2n−1)α≤an (x)<(2n)α,1≤n≤N
|IN (a1, . . . , an)|s
≤ liminf
N→∞
∑
a1,...,aN :(2n−1)α≤an<(2n)α,1≤n≤N
N∏
n=1
1
a2sn
= liminf
N→∞
N∏
n=1
∑
(2n−1)α≤an<(2n)α
1
a2sn
.
For any s > α−12α , one can choose no such that for all n ≥ no ,∑
(2n−1)α≤an<(2n)α
1
a2sn
≤ (2n)
α− (2n −1)α
(2n −1)2αs ≤
2α · (2n)α−1
(2n −1)2αs < 1,
and thus it follows that
H s (Eα) ≤
no∏
n=1
∑
(2n−1)α≤an<(2n)α
1
a2sn
<∞.
As a result,
dimH Eα ≤ α−1
2α
.
(II) For the lower bound of dimH Eα, notice that the set Eα has a nice Cantor structure. For
each n ≥ 1, let
En =
{
In(a1, . . . , an) : (2k −1)α ≤ ak < (2k)α, for all 1 ≤ k ≤ n
}
.
Then
E =
∞⋂
n=1
⋃
In (a1,...,an )∈En
In(a1, . . . , an),
and every element in In−1(a1, . . . , an−1) in En−1 contains exactly
c1n
α−1 ≤ Dn := (2n)α− (2n −1)α ≤ c2nα−1.
elements In(a1, . . . , an) in En .
Then, we distribute a mass supported on Eα by setting
µ(In(a1, . . . , an)) = 1
D1 · · ·Dn
,
for all n ≥ 1 and In(a1, . . . , an) in En .
For any x ∈ Eα, for each r > 0 small enough, let n be the integer such that
|In+1(x)| < r ≤ |In(x)|.
Then, by Proposition 3, the ball B(x,r ) can intersect at most three cylinders of order n,
thus
liminf
r→∞
logµ(B(x,r ))
logr
≥ liminf
n→∞
log[3 · (D1 · · ·Dn)−1]
log |In+1(x)|
.
By the recursive relation of qn(x), it follows that
qn+1(x) ≤
n+1∏
k=1
(ak (x)+1) ≤
n+1∏
k=1
(2k)α ≤ cn
n+1∏
k=1
kα;
on the other hand,
n∏
k=1
Dk ≥ cn1
n∏
k=1
kα−1.
1094 Xuan Zhao and Zhenliang Zhang
Thus it follows that
liminf
r→0
logµ((x,r ))
logr
≥ liminf
n→∞
log
∏n
k=1 k
α−1
log
(∏n+1
k=1 k
α
)2 = α−12α .
Finally by Proposition 4, one has
dimH Eα ≥ α−1
2α
. □
By a result in [5] or the same line of the argument as above, except minor modifications on
notation, one can show the following.
Lemma 6. Let b > 1. Define
Fb =
{
x ∈ [0,1) : b1+2+···+n ≤ an(x) < 2 ·b1+2+···+n , for all n ≥ 1
}
.
Then dimH Fb = 1/2.
Proof of Theorem 1. It is clear that for any α> 1,
Eα ⊂SG , and so, dimH SG ≥ lim
α→∞
α−1
2α
= 1
2
.
It is also clear that for any b > 1,
Fb ⊂ EG , and so, dimH EG ≥
1
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.
References
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theory of continued fractions”, Nonlinearity 33 (2020), no. 6, pp. 2615–2639.
[2] A. Bakhtawar, P. Bos and M. Hussain, “The sets of Dirichlet non-improvable numbers
versus well-approximable numbers”, Ergodic Theory Dyn. Syst. 40 (2020), no. 12, pp. 3217–
3235.
[3] P. Bos, M. Hussain and D. Simmons, “The generalised Hausdorffmeasure of sets of Dirich-
let non-improvable numbers”, Proc. Am. Math. Soc. 151 (2023), no. 5, pp. 1823–1838.
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Xuan Zhao and Zhenliang Zhang 1095
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Some lacunarity properties of partial quotients of real numbers

Nicolas Curien

Lucile Laulin

Comptes Rendus
Mathématique
Nicolas Curien and Lucile Laulin
Recurrence of the plane Elephant random walk
Volume 362 (2024), p. 1183-1188
Online since: 5 November 2024
https://doi.org/10.5802/crmath.620
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. 1183-1188
https://doi.org/10.5802/crmath.620
Research article / Article de recherche
Probability theory / Probabilités
Recurrence of the plane Elephant random
walk
Récurrence de la marche aléatoire de l’éléphant dans le
plan
Nicolas Curien a and Lucile Laulin ∗, ,b
a Laboratoire de mathématiques d’Orsay – UMR CNRS 8628, Université Paris-Saclay,
91405 Orsay, France
b Modal’X - UMR CNRS 9023, Université Paris Nanterre, 92000 Nanterre, France
E-mails: nicolas.curien@gmail.com, lucile.laulin@math.cnrs.fr
Abstract. We give a short proof of the recurrence of the two-dimensional elephant random walk in the diffu-
sive regime. This was recently established by Qin (2023), but our proof mainly uses very rough comparison
with the standard plane random walk. We hope that the method can be useful for other applications.
Résumé. Nous donnons une preuve courte de la récurrence de la marche aléatoire de l’éléphant dans le plan
dans le régime diffusif. Cela a récemment été établi par Shuo Qin, mais notre preuve ne repose que sur une
comparaison avec la marche aléatoire simple dans le plan. Nous espérons que cette méthode puisse être utile
pour d’autres applications.
Keywords. Elephant random walk, recurrence of random walk, martingales.
Mots-clés. Marche aléatoire de l’éléphant, Récurrence de la marche aléatoire, Martingales.
2020 Mathematics Subject Classification. 60G50, 60G42, 60J10, 82C41.
Manuscript received 23 October 2023, accepted 30 January 2024.
1. Introduction
The elephant random walk on Zd has been introduced in dimension 1 by Schütz and Trimper [6]
and is a well-studied discrete process with reinforcement, see [3] for background and references.
Its definition (see (2.1)) depends on a memory parameter1 α ∈ (− 12d−1 ,1) and it exhibits a phase
transition going from a diffusive when α< αc = 12 to a superdiffusive behavior when α> αc . We
focus here on the two-dimensional case and establish recurrence of the process in the diffusive
regime.
Theorem 1. In the diffusive regime α<αc = 12 , the plane elephant random walk is recurrent.
∗Corresponding author
1The usual definition uses a memory parameter p ∈ [0,1] which is the probability to reproduce a (uniform) former
step of the walk, or to move in one of the 3 remaining directions with the same probability (1−p)/3 so that α= (4p−1)/3,
see [3, Equation (1.4)].
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1184 Nicolas Curien and Lucile Laulin
This has been recently proved by Qin [5] but our approach is different and much shorter,
however it gives less quantitive information and does not directly apply in the critical regime
α = αc . We use a comparaison to the simple random walk which could apply in dimension 1 as
well since the simple random walk is also recurrent in that case.
It is worth pointing out that [5] established that the elephant random walk is always transient
when d ≥ 3, similar to the simple random walk.
Notation. We write ei the four directions of Z2 for 1 ≤ i ≤ 4. We shall write (Xk : k ≥ 0) for
the canonical underlying process starting from 0 := (0,0) ∈ Z2, we denote its steps by ∆Xk =
Xk+1−Xk ∈ {e1,e2,e3,e4} and we introduce for 1 ≤ i ≤ 4 the centered counting direction processes
D [X ]k (ei ) defined by
D [X ]k (ei ) =
k−1∑
j=0
1{X j+1−X j =ei } −
k
4
, in particular notice that
4∑
i=1
D [X ]n (ei ) = 0. (1.1)
For any stopping time θ, we denote by X (θ) the shifted process X (θ)k = Xθ+k −Xθ for k ≥ 0. Finally
Fn is the canonical filtration generated by the first n steps of the walk and we use X[0,n] as a
shorthand for (Xk : 0 ≤ k ≤ n).
2. Comparison between elephant and simple random walk
Under the law P the underlying process (X ) evolves as the standard simple random walk on Z2,
whereas under P , it evolves as the α-elephant random walk i.e. satisfying for n ≥ 0,
P (∆Xn = ei |Fn) = 1
4
+αD
[X ]
n (ei )
n
, (2.1)
(where we interpret 0/0 = 0 for n = 0). In particular, under P , the process (D [X ]k (ei ) : 1 ≤ i ≤ 4,
k ≥ 0) is Markov and evolves as an urn process with four colors, which was crucially used in [1]
to establish the phase transition diffusive/superdiffusive. The local evolution of the elephant
random walk (for large times) ressembles that of the simple random walk and this is quantified
in the following proposition:
Proposition 2 (Markov contiguity). For any ε > 0 and any A > 0, there exist cε,A > 0 and a
sequence of events En satisfying liminfn→∞P (X[0,n] ∈ En) ≥ 1− ε such that for any measurable
function f ,
E
[
f
(
X (n)[0,n]
)
1X (n)[0,n]∈En
∣∣∣∣∣ Fn and |D [X ]n (ei )|pn ≤ A for all 1 ≤ i ≤ 4
]
≥ cε,A ·E
[
f
(
X[0,n])1X[0,n]∈En
]
.
Proof. In the event considered in the conditioning, we have |D [X ]n (ei )| ≤ A
p
n for all 1 ≤ i ≤ 4.
By (2.1), the Radon–Nikodym derivative of (Xn+k −Xn : 0 ≤ k ≤ n) under P with respect to P is
given by
RNDn :=
2n−1∏
k=n
(
1+α
D [X ]k (∆Xk )
k
)
=
n−1∏
k=0
1+αD [X ]n (∆X (n)k )+D [X (n)]k (∆X (n)k )
n +k
 . (2.2)
By Donsker’s invariance principle, we can find a constant Aε such that the event
Gn =
{
max
i
sup
0≤k≤n
∣∣∣D [X (n)]k (ei )∣∣∣≤ Aεpn}
has probability at least 1 − ε under P . On this event (and conditionally on the event of the
statement of the proposition), the counting directions processes D [X ][n,2n](ei ) are in absolute value
Nicolas Curien and Lucile Laulin 1185
bounded by (A+Aε)
p
n. In particular, using log(1+αx) ≥αx−(αx)2 for small |x|, we deduce that
on this event, for n large enough, the Radon–Nikodym derivative in (2.2) is lower bounded by
RNDn 1Gn ≥ exp
(
αMn −α2(A+ Aε)2
)
where M j =
j−1∑
k=0
D [X ]n+k (∆Xn+k )
n +k .
Using (1.1), it is trivial to check that (M j : 0 ≤ j ≤ n) is a (Fn+·)- martingale with quadratic
variation
E
[
M 2j+1 −M 2j
∣∣∣ Fn+ j ]= E[(M j+1 −M j )2∣∣Fn+ j ]= 1
4
∑4
i=1
(
D [X ]n+ j (ei )
)2
(n + j )2 ≤on Gn
(A+ Aε)2
4n
.
It follows that E[M 2n 1Gn ] ≤ (A+Aε)
2
4 . In particular, thanks to Markov inequality, for any ε > 0, the
event Hn =
{|Mn |1Gn < (A+Aε)2pε } is of probability at least 1− ε. Gathering up the pieces, on the
event En =Gn ∩Hn which is of P measure at least 1−2ε, the Radon–Nikodym derivative of the
elephant w.r.t. the simple random walk is at least e
−α (A+Aε)
2
p
ε
−α2(A+Aε)2 =: cε,A . □
Xn
0
X2n
X
(n)
[0,n]
X
(2n)
[0,n]
Figure 1. Illustration of the proof of Proposition 3. Conditionally on Fn and on the fact
that the counting directions processes are controlled at time n, the blue and red parts are
independent on events of large probability. This is sufficient to imply a lower bound on the
probability of return to 0.
It is classical that in the plane, the simple random walk started from x ∈ Z2 with ∥x∥ ≈ pn
has a probability of order log−1 n to visit (0,0) within n steps. Our weak bound (Proposition 2) is
sufficient to imply the same kind of estimate for the elephant random walk:
Proposition 3. For any A > 0 there exists cA > 0 such that
E
[
∃ 5
2
n ≤ k ≤ 3n : Xk = 0
∣∣∣∣∣ Fn and |D [X ]n (ei )|pn ≤ A for all 1 ≤ i ≤ 4
]
≥ cA
logn
.
1186 Nicolas Curien and Lucile Laulin
Proof. Let us denote xn = Xn which is fixed conditionally on Fn . Using Proposition 2 twice, for
any positive functions f and g and any A, A′ > 0 and any ε > 0, we can find two sequences of
events En and E ′n and constants cε,A and cε,A′ such that
E
[
f (X (2n)[0,n])g (X
(n)
[0,n])
∣∣∣Fn]
≥ E
[
f (X (2n)[0,n])g (X
(n)
[0,n])1 ∥D[X ]2n (ei )∥p
n
≤A′, ∀1≤i≤4
∣∣∣∣∣ Fn
]
≥ cε,A′ ·E
[
f (X[0,n])1X[0,n]∈E ′n
]
·E
[
1 ∥D[X ]2n (ei )∥p
n
≤A′, ∀1≤i≤4
g (X (n)[0,n])
∣∣∣∣∣ Fn
]
≥ cε,A′ ·E
[
f (X[0,n])1X[0,n]∈E ′n
]
· cε,A ·E
[
g (X[0,n])1X[0,n]∈En 1 ∥D[X ]n (ei )∥p
n
≤A′−A, ∀1≤i≤4
]
1 ∥D[X ]n (ei )∥p
n
≤A, ∀1≤i≤4
.
Up to increasing A′ we may suppose that the event Hn = En ∩E ′n ∩
{ ∥D [X ]n (ei )∥p
n
≤ A′−A, ∀ 1 ≤ i ≤ 4
}
has probability at least 1−3ε and particularizing the inequality above, we deduce that for some
constant c̃ε,A > 0 the probability in the proposition is lower bounded by
c̃ε,A ·P
(
∃ 3
2
n ≤ k ≤ 2n : Xk =−xn and
X (0)[0,n] ∈ Hn
X (n)[0,n] ∈ Hn
)
,
so that we can apply the following lemma to conclude.
Lemma 4. For any A > 0, there exists ε > 0 and δA > 0 so that if xn ∈ Z2 is such that ∥xn∥ ≤ A
p
n
and if En is a sequence of events such that P (X[0,n] ∈ En) ≥ 1−ε then we have
P
(
∃ 3
2
n ≤ k ≤ 2n : Xk =−xn and
X (0)[0,n] ∈ En
X (n)[0,n] ∈ En
)
≥ δA
logn
.
Proof. We use a second-moment method on the random variable
N
En
xn := #
{
3
2
n ≤ k ≤ 2n : Xk =−xn
}
1X (0)[0,n]∈En
1X (n)[0,n]∈En
.
We denote by pEnk (y) = E [1Xk=y 1X[0,n]∈En ] and pk (y) = P (Xk = y) for the heat kernels. By the
standard local limit theorem (or just Stirling approximation on the binomial coefficients) there
exists C > 0 such that pk (y) ≤ Ck for all k ≥ 1 and y ∈Z2. First, by lifting the restrictions on En we
have
E
[(
N
En
xn
)2]≤ E [( 2n∑
k=3/2n
1Xk=−xn
)2]
≤ 2 ∑
3
2 n≤k≤k ′≤2n
pk (−xn)pk ′−k (0)
≤ 2 ∑
3
2 n≤k≤k ′≤2n
C
n
C
k ′−k ≤C
′ log(n),
for some C ′ > 0 (independent of n). To evaluate the first moment, introduce the (truncated)
Green functions g En (y) = ∑nk=n/2 pEnk (y) and similarly g (y) = ∑nk=n/2 pk (y). In particular, since
Nicolas Curien and Lucile Laulin 1187
P (En) ≥ 1 − ε we have ∥p − pEn∥1 := ∑y p(y) − pEn (y) ≤ ε and similarly and ∥g − g En∥1 =∑
y g (y)− g En (y) ≤ εn. Recalling that Cn ≥ pEnn (y) ≥ pn(y) and 2C ≥ g En (y) ≥ g (y), we have
E
[
N
En
xn
]
= ∑
y∈Z2
pEnn (y)g
En (−y −xn) =
∑
y∈Z2
 p
En
n (y)g
En (−y −xn)−pEnn (y)g (−y −xn)
−pn(y)g (−y −xn)+pEnn (y)g (−y −xn)
+pn(y)g (−y −xn)

≥∑
y
pn(y)g (−y −xn)−
∥∥∥pEnn ∥∥∥∞∥∥g − g En ∥∥1 −∥g∥∞∥∥∥pn −pEnn ∥∥∥
≥∑
y
pn(y)g (−y −xn)−3C 2ε.
However, since ∥xn∥ ≤ A
p
n, the local limit theorem implies that
∑
y p(y)g (−y−xn) > cA for some
cA > 0 independently of n and so one can choose ε> 0 small enough so that if P (En) ≥ 1−ε then
we have E
[
N
En
xn
]> cA/2. We conclude by the second moment method that
P (N Enxn > 0) ≥ E
[
N
En
xn
]
/E
[(
N
En
xn
)2]≥ cA
2C ′ logn
. □
3. Proof of Theorem 1
Proof of Theorem 1. Let us denote P3 j =P (∃ 3 j ≤ k ≤ 3 j+1, Xk = 0 |F3 j ). When α<αc , i.e. the
diffusive regime, Bertenghi [1, Theorem 4.2] showed that under P we have(
D Xn (ei )p
n
)
1≤i≤4
(d)−−−−→
n→∞ (Xi )1≤i≤4,
for some random variable X (whose distribution is irrelevant for our purposes). Together with
our Proposition 3, this shows that in the diffusive regime, for any ε> 0 there exists δ> 0 such that
for large j ’s we have
P ( j ·P3 j > δ) ≥ 1−ε. (3.1)
Notice that the variables P3 j are not independent, but Jeulin’s lemma [4, Proposition 3.2] gives
∞∑
k=1
P3k =∞, P −a.s. (3.2)
To be honest we rather use the proof than the lemma itself, and since the argument is short
let us reproduce it here: Suppose by contradiction that there exists ε, M > 0 so that the event
A = {∑∞k=1 P3k < M} has probability at least ε> 0. Using (3.1) we take δ> 0 so that P ( j ·P3 j >
δ) ≥ 1− ε2 and write
M ≥ E
[
1A
∑
j≥1
P3k
]
= ∑
j≥1
δ
j
·P
(
A∩
{
P3 j >
δ
j
})
︸ ︷︷ ︸
≥ε−(1−(1− ε2 ))=ε/2
=∞,
which is a contradiction. Given (3.2), the conditional Borel–Cantelli lemma ([2, Theorem 4.3.4])
then implies that the events {∃ 3 j ≤ k ≤ 3 j+1 : Xk = 0} happen for infinitely many j ’s with
probability one, implying recurrence of the process. □
Acknowledgments
We thank Jean Bertoin and Thomas Budzinski for discussions about the plane elephant random
walk. We are also indebted to Alice Contat for providing us with the pictures and .
1188 Nicolas Curien and Lucile Laulin
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] M. Bertenghi, “Functional limit theorems for the multi-dimensional elephant random
walk”, Stoch. Models 38 (2022), no. 1, pp. 37–50.
[2] R. Durrett, Probability. Theory and examples, Fifth edition, Cambridge University Press,
2019, pp. xii+419.
[3] L. Laulin, About the elephant random walk, PhD thesis, Université de Bordeaux, 2022.
[4] A. Matsumoto and K. Yano, “On a zero-one law for the norm process of transient random
walk”, in Séminaire de Probabilités XLIII (C. Donati-Martin, A. Lejay and A. Rouault, eds.),
Springer, 2011, pp. 105–126.
[5] S. Qin, “Recurrence and transience of multidimensional elephant random walks”, 2023.
https://arxiv.org/abs/2309.09795.
[6] G. M. Schütz and S. Trimper, “Elephants can always remember: Exact long-range memory
effects in a non-Markovian random walk”, Phys. Rev. E 70 (2004), article no. 045101 (4 pages).


Recurrence of the plane Elephant random walk

Boris Adamczewski

Colin Faverjon

Relations algébriques entre valeurs de $E$-fonctions ou de $M$-fonctions

Junpeng Wang

Zhongkui Liu

Renyu Zhao

When are the classes of Gorenstein modules (co)tilting?

María Cumplido

Federica Gavazzi

Luis Paris

Comptes Rendus
Mathématique
María Cumplido, Federica Gavazzi and Luis Paris
Intersection of parabolic subgroups in Euclidean braid groups: a short proof
Volume 362 (2024), p. 1445-1448
Online since: 14 November 2024
https://doi.org/10.5802/crmath.656
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. 1445-1448
https://doi.org/10.5802/crmath.656
Research article / Article de recherche
Group theory / Théorie des groupes
Intersection of parabolic subgroups in
Euclidean braid groups: a short proof
Intersection de sous-groupes paraboliques dans les
groupes de tresses euclidiens : une démonstration courte
María Cumplido ∗,a, Federica Gavazzi b and Luis Paris b
a Departamento de Álgebra, Facultad de Matemáticas, Universidad de Sevilla, Calle
Tarfia s/n 41012, Seville, Spain
b IMB, UMR5584, CNRS, Université de Bourgogne, 21000 Dijon, France
E-mails: cumplido@us.es, federica.gavazzi@u-bourgogne.fr,
luis.paris@u-bourgogne.fr
Abstract. We give a short proof for the fact, already proven by Thomas Haettel, that the arbitrary intersection
of parabolic subgroups in Euclidean Braid groups A[Ãn ] is again a parabolic subgroup. To that end, we use
that the spherical-type Artin group A[Bn+1] is isomorphic to A[Ãn ]⋊Z .
Résumé. Nous donnons une démonstration courte du fait, déjà démontré par Thomas Haettel, que l’inter-
section arbitraire de sous-groupes paraboliques dans les groupes de tresses euclidiens A[Ãn ] est à nouveau
un sous-groupe parabolique. À cette fin, nous utilisons le fait que le groupe d’Artin de type sphérique A[Bn+1]
est isomorphe à A[Ãn ]⋊Z.
Keywords. Group theory, Artin groups, Euclidean braid groups, parabolic subgroups, group isomorphism.
Mots-clés. Théorie des groupes, groupes d’Artin, groupes de tresses euclidiens, sous-groupes paraboliques,
isomorphisme de groupes.
2020 Mathematics Subject Classification. 20F36.
Funding. The first author is supported by a Ramón y Cajal 2021 grant and the research project PID2022-
138719NA-100, financed by MCIN/AEI/10.13039/501100011033/FEDER, UE.
Manuscript received 24 January 2024, revised 23 May 2024, accepted 3 June 2024.
An Artin (or Artin-Tits) group AS is any group with a presentation
〈S| st s · · ·︸ ︷︷ ︸
mst
= t st · · ·︸ ︷︷ ︸
mst
, s, t ∈ S, ms,t ̸=∞, s ̸= t〉
where S is a finite set of generators and (ms,t )s,t∈S is a symmetric matrix with 1’s in the diagonal
and the other entrances in {2,3, . . . ,∞}. The number of known global results for these groups
is very limited, and for some decades now, classic problems such as the word problem, the
conjugacy problem, or the K (π,1) conjecture have been the subject of study by group theorists.
Specifically, it has become necessary to better study the properties of certain specific subgroups:
the parabolic subgroups. A standard parabolic subgroup AX of AS is a subgroup generated
by a subset X ⊂ S, and thanks to [8], we also know that it coincides with the Artin group
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1446 María Cumplido, Federica Gavazzi and Luis Paris
on X with the same relations that these generators have in AS . A parabolic subgroup is any
conjugate of a standard parabolic subgroup. These subgroups play a principal role in the
construction of simplicial complexes associated with Artin groups [1–4, 7], either as stabilizers
of simplices or as the building blocks of the complexes. In the case of braid groups, the parabolic
subgroups coincide with the isotopy classes of non-degenerate multicurves in the n-punctured
disc (containing the vertices of the same curve complex). However, the basic question of whether
the arbitrary intersection of parabolic subgroups is a parabolic subgroup remains open in most
cases.
To establish notations, we will use Coxeter graphs. A Coxeter graph ΓS encodes the informa-
tion of an Artin group AS as follows: each generator corresponds to a vertex, and two vertices s, t
are connected by an edge if the vertices do not commute. This edge is labeled by ms,t if ms,t > 3
and by ∞ if there is no relation between s and t . In this way, we can also refer to AS as A[ΓS ].
The Euclidean braid group (also known as affine braid group) with n +1 generators is the group
A[Ãn], where Ãn is the graph in Figure 1:
Figure 1. The Coxeter graph Ãn .
In this article, we will give an alternative and concise proof of the following theorem:
Theorem 1 ([5, Corollary N]). Any intersection of parabolic subgroups in A[Ãn] is a parabolic
subgroup.
We know, according to [2], that the intersection of parabolic subgroups of a spherical-type
Artin group remains a parabolic subgroup. In particular, this is true in the group A[Bn+1], where
Bn+1 is the graph illustrated in Figure 2.
Figure 2. The Coxeter graph Bn+1.
Using the notation of the figure, let ρ = r1 . . .rnrn+1. For 1 ≤ i ≤ n−1, it holds that ρriρ−1 = ri+1.
If we additionally define r0 := ρrnρ−1, we can state that the previous equality is true modulo
n +1 (observing that ρ2rnρ−2 = r1 is sufficient). Furthermore, we consider the following exterior
automorphism f of A[Ãn]:
f : A[Ãn] −→ A[Ãn]
ti 7−→ ti+1,
again with the indexes taken modulo n+1. Given this, we can define an action of the infinite cyclic
groupZ∼= 〈u〉 on A[Ãn] by setting u ·g = ug u−1 := f (g ), for all g ∈ A[Ãn]. With this action, we can
define the semidirect product A[Ãn]⋊ 〈u〉, that has a presentation with generators {t1, . . . , tn ,u}
and relations given by the union of the Artin relations of A[Ãn] and the set {uti u−1 = ti+1 |
0 ≤ i ≤ n} modulo n +1.
María Cumplido, Federica Gavazzi and Luis Paris 1447
Theorem 2 ([6]). The map ϕ : A[Ãn] ⋊ 〈u〉 −→ A[Bn+1] that sends ti to ri and u to ρ is an
isomorphism.
Remark 3. The restriction of ϕ to A[Ãn] gives an embedding of this Artin group in A[Bn+1].
The proof of the next result is a straightforward consequence of the definition of ϕ.
Lemma 4. Let ξ be the group homomorphism defined by
ξ : A[Bn+1] −→Z
ri 7−→ 0 for 1 ≤ i ≤ n,
rn+1 7−→ 1.
Then ρ is mapped to 1, and the kernel of ξ is ϕ(A[Ãn]).
Proof of Theorem 1. Firstly, notice that if P is a proper parabolic subgroup of A[Ãn], thenϕ(P ) is
a parabolic subgroup of A[Bn+1]. The only case in which this is not clear is when P = g A[Ãn]X g−1
and X contains t0, which is sent to ρrnρ−1 by ϕ. However, since P is proper, X does not contain
all the generators of A[Ãn], thus, using ρ, we can always conjugate ϕ(X ) to a subset of {r1, . . . ,rn}.
Now suppose that P1 and P2 are two parabolic subgroups of A[Ãn]. Since in A[Bn+1] the
intersection of parabolic subgroups is a parabolic subgroup, we have, in particular, that ϕ(P1 ∩
P2) =ϕ(P1)∩ϕ(P2) is a parabolic subgroup of A[Bn+1].
To complete the proof, it remains to show that if Q is a parabolic subgroup of A[Bn+1]
such that P := ϕ−1(Q) ⊂ A[Ãn], then P is a parabolic subgroup of A[Ãn]. We can write Q =
h A[Bn+1]Y h−1, with Y ⊂ {r1, . . . ,rn+1} and h ∈ A[Bn+1]. First, we show that rn+1 ̸∈ Y . If we suppose
otherwise, then hrn+1h−1 ∈ Q and ϕ−1(hrn+1h−1) ∈ P ⊂ A[Ãn]. In particular, by Lemma 4, the
element hrn+1h−1 must belong to the kernel of ξ, that coincides with ϕ(A[Ãn]). But computing
ξ(hrn+1h−1) = 1, we get a contradiction. Therefore, Y ⊂ {r1, . . . ,rn} =ϕ({t1, . . . , tn}). Furthermore,
since A[Ãn]⋊ 〈u〉 ∼= A[Bn+1], we can write h = h1ρm , with ϕ−1(h1) ∈ A[Ãn] and m ∈ Z. Thus,
Q = h1ρm A[Bn+1]Y ρ−mh−11 , and exploiting the fact that um A[Ãn]ϕ−1(Y )u−m = f m(A[Ãn]ϕ−1(Y )),
we get
P =ϕ−1(Q) =ϕ−1(h1) f m(A[Ãn]ϕ−1(Y ))(ϕ−1(h1))−1 =ϕ−1(h1)A[Ãn] f m (ϕ−1(Y ))(ϕ−1(h1))−1,
which has the form of a parabolic subgroup in A[Ãn]. Finally (see the complete argument
in [3, Corollary 16]), in any Artin group, any descending chain of inclusions of parabolic sub-
groups has to stabilize, and then the finite intersection of parabolic subgroups, being a parabolic
subgroup, implies that the same is true for an arbitrary intersection. □
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] R. Charney and M. W. Davis, “The K (π,1)-problem for hyperplane complements associated
to infinite reflection groups”, J. Am. Math. Soc. 8 (1995), no. 3, pp. 597–627.
[2] M. Cumplido, V. Gebhardt, J. González-Meneses and B. Wiest, “On parabolic subgroups of
Artin–Tits groups of spherical type”, Adv. Math. 352 (2019), pp. 572–610.
[3] M. Cumplido, A. Martin and N. Vaskou, “Parabolic subgroups of large-type Artin groups”,
Math. Proc. Camb. Philos. Soc. 174 (2023), no. 2, pp. 393–414.
1448 María Cumplido, Federica Gavazzi and Luis Paris
[4] C. De Concini and M. Salvetti, “Cohomology of Coxeter groups and Artin groups”, Math. Res.
Lett. 7 (2000), no. 2-3, pp. 213–232.
[5] T. Haettel, “Lattices, injective metrics and the K (π,1) conjecture”, 2021. to appear in Alge-
braic & Geometric Topology, https://arxiv.org/abs/2109.07891.
[6] R. P. Kent IV and D. Peifer, “A geometric and algebraic description of annular braid groups”,
Int. J. Algebra Comput. 12 (2002), no. 1-2, pp. 85–97.
[7] L. Paris, “K (π,1) conjecture for Artin groups”, Ann. Fac. Sci. Toulouse, Math. 23 (2014), no. 2,
pp. 361–415.
[8] H. Van der Lek, The Homotopy Type of Complex Hyperplane Complements, PhD thesis,
Nijmegen, 1983.


Intersection of parabolic subgroups in Euclidean braid groups: a short proof

Russelle Guadalupe

Comptes Rendus
Mathématique
Russelle Guadalupe
A note on the exact formulas for certain 2-color partitions
Volume 362 (2024), p. 1485-1490
Online since: 14 November 2024
https://doi.org/10.5802/crmath.658
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. 1485-1490
https://doi.org/10.5802/crmath.658
Research article / Article de recherche
Number theory / Théorie des nombres
A note on the exact formulas for certain
2-color partitions
Note sur les formules exactes de certaines partitions à 2
couleurs
Russelle Guadalupe ,a
a Institute of Mathematics, University of the Philippines-Diliman, Quezon City, 1101, Philippines
E-mail: rguadalupe@math.upd.edu.ph
Abstract. Let p ≤ 23 be a prime and ap (n) count the number of partitions of n where parts that are multiple
of p come up with 2 colors. Using a result of Sussman, we derive the exact formula for ap (n) and obtain an
asymptotic formula for log ap (n). Our results partially extend the work of Mauth, who proved the asymptotic
formula for log a2(n) conjectured by Banerjee et al.
Résumé. Soit p ≤ 23 un nombre premier et ap (n) compte le nombre de partitions de n où les parties qui sont
multiples de p donnent 2 couleurs. En utilisant un résultat de Sussman, nous dérivons la formule exacte pour
ap (n) et obtenons une formule asymptotique pour log ap (n). Nos résultats étendent partiellement le travail
de Mauth, qui a prouvé la formule asymptotique pour log a2(n) conjecturée par Banerjee et al.
Keywords. Circle method, η-quotients, partitions, asymptotic formula.
Mots-clés. Méthode des cercles, η-quotients, partitions, formule asymptotique.
2020 Mathematics Subject Classification. 11P55, 11P82, 05A16.
Manuscript received 2 April 2024, revised 23 May 2024, accepted 17 June 2024.
1. Introduction
Throughout this paper, we denote (a; q)∞ = ∏n≥0(1− aqn) for a ∈ C. Recall that a partition of
a positive integer n is a nonincreasing finite sequence of positive integers, known as its parts,
whose sum is n. We define p(n) as the number of partitions of n, which can be seen as the
coefficients of its generating function given by
1
(q ; q)∞
=
∞∑
n=0
p(n)qn .
Hardy and Ramanujan [9] proved the following asymptotic formula for p(n) given by
p(n) ∼ 1
4n
p
3
exp
(
π
√
2n
3
)
, n −→∞,
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1486 Russelle Guadalupe
using the celebrated Circle Method. Rademacher [14] refined the Hardy–Ramanujan Circle
Method and derived the exact formula
p(n) = 1
π
p
2
∞∑
k=1
Ak (n)
p
k
d
dn
sinh
(
π
k
√
2
3
(
n − 124
))
√
n − 124

where
Ak (n) :=
∑
0≤h<k
gcd(h,k)=1
exp
[
πi
(
s(h,k)− 2nh
k
)]
and
s(h,k) :=
k−1∑
j=1
j
k
({
h j
k
}
− 1
2
)
is the Dedekind sum, where {t } denotes the fractional part of t . Recently, Banerjee et al. [3] proved
the following refined asymptotic formula for p(n) given by
log p(n) ∼π
√
2n
3
− logn − log4p3− 0.44 · · ·p
n
, n −→∞,
from a family of inequalities for p(n), which was used to prove a refinement of the inequality of
DeSalvo and Pak [8], and Chen, Wang and Xie [6] that reads(
1+ π
24n3/2
− 1
n2
)
p(n −1)p(n +1) < p(n)2 <
(
1+ π
24n3/2
)
p(n −1)p(n +1)
for n ≥ 120.
Let ak (n) be the number of partitions of n where parts that are multiple of k come up with
2 colors. For k = 1, we have a1(n) = p2(n) where p2(n) is the total number of partitions of n
where parts come up with 2 colors and for k = 2, a2(n) also counts the number of cubic partitions
(see [10]). The generating function for ak (n) is given by [1]
1
(q ; q)∞(qk ; qk )∞
=
∞∑
n=0
ak (n)q
n .
Kotesovec [11] found the following asymptotic formula for a2(n) given by
a2(n) ∼ 1
8n5/4
exp(π
p
n), n −→∞.
Banerjee et al. [3] conjectured that a2(n) satisfies the following asymptotic formula
log a2(n) ∼π
p
n − 5
4
logn − log8− 0.79 · · ·p
n
, n −→∞. (1)
Recently, Mauth [12] proved (1) by finding the exact formula of Rademacher type for a2(n) using
a result of Zuckerman [16].
In this paper, we use a result of Sussman [15] to derive Rademacher-type formulas for ap (n)
when p ≤ 23 is a prime, and deduce their asymptotic formulas in the spirit of (1), which can be
considered as a partial extension of Mauth’s work. We give our main results as follows.
Theorem 1. Let p ≤ 23 be a prime. Then for n ≥ 1 we have
ap (n) = 2πpp
(
1+p−1
24n −p −1
)p−1∑
j=1
∞∑
m=1
m≡ j mod p
I2
(
π
6m
√
(1+p−1)(24n −p −1)
)
bm(n)
m
+2π
(
1+p
24n −p −1
) ∞∑
m=1
p|m
I2
( π
6m
√
(1+p)(24n −p −1)
) bm(n)
m
,
Russelle Guadalupe 1487
where
bk (n) :=
∑
0≤h<k
gcd(h,k)=1
exp
[
−2πnhi
k
+πs(h,k)i+πs
(
ph
gcd(p,k)
,
k
gcd(p,k)
)
i
]
and I2(s) is the second modified Bessel function of the first kind (see Section 2).
Corollary 2. For p ≤ 23 prime, we have
log ap (n) ∼π
√
2n(1+p−1)
3
− 5
4
logn + log 2
√
3p(1+p−1)3/4
245/4
− cp
24
p
6n
, n −→∞
where
cp := 135
π
√
1+p−1
+π
√
(1+p)3
p
.
Setting p = 2 in Corollary 2, we deduce the asymptotic formula due to Mauth [12] given by
log a2(n) ∼π
p
n − 5
4
logn − log8−
(
15
8π
+ π
16
)
1p
n
, n −→∞,
which immediately yields (1).
The paper is organized as follows. In Section 2, we state the Sussman’s result on the exact
formulas for the Fourier coefficients of a class of η-quotients. In Section 3, we apply this result
to prove Theorem 1, and using the asymptotic expansion of I2(s) due to Banerjee [2], we deduce
Corollary 2.
2. Fourier coefficients of a class of η-quotients
We consider in this section the exact formulas for the Fourier coefficients of the following class of
holomorphic functions on the open disk given by
G(q) :=
R∏
r=1
(qmr ; qmr )δr∞ =
∞∑
n=0
g (n)qn ,
where m = (m1, . . . ,mR ) is a sequence of R distinct positive integers and δ = (δ1, . . . ,δR ) is a
sequence of R nonzero integers. The functions G(q) can be seen as η-quotients since (q ; q)∞ =
q−1/24η(τ), where Dedekind’s eta function, denoted by η(τ), is defined by the infinite product
η(τ) := q1/24 ∏n≥1(1−qn) with q = e2πiτand τ ∈H := {z ∈C : Im(z) > 0}. For useful properties of the
Dedekind’s eta function and the definition of η-quotients, we refer the interested reader to the
book [13]. Given a particular G(q) with
∑R
r=1δr < 0, Sussman [15] gave a Rademacher-type exact
formula for g (n), which is a special case of the work of Bringmann and Ono [4] on the coefficients
of harmonic Maass forms. Sussman’s proof follows the original approach of Rademacher [14]
on the Hardy–Ramanujan Circle Method. In the case where
∑R
r=1δr ≥ 0, Chern [7] obtained an
analogous formula with an error term using the method of O-Y. Chan [5].
Before we state Sussman’s result, we need some definitions. For (h,k) ∈N2 with gcd(h,k) = 1,
we set
∆1 =−1
2
R∑
r=1
δr , ∆2 =
R∑
r=1
mrδr ,
∆3(k) =−
R∑
r=1
gcd(mr ,k)2
mr
δr , ∆4(k) =
R∏
r=1
(
gcd(mr ,k)
mr
)δr /2
,
1488 Russelle Guadalupe
and
Âk (n) =
∑
0≤h<k
gcd(h,k)=1
exp
(
−2πnhi
k
−πi
R∑
r=1
δr s
(
mr h
gcd(mr ,k)
,
k
gcd(mr ,k)
))
,
where s(h,k) is the Dedekind sum defined in Section 1. We also set L as the least common
multiple of m1, . . . ,mR , and partition the set {1, . . . ,L} into two disjoint subsets
L>0 := {1 ≤ l ≤ L :∆3(l ) > 0},
L≤0 := {1 ≤ l ≤ L :∆3(l ) ≤ 0}.
Theorem 3 ([15]). If ∆1 > 0 and the inequality
min
1≤r≤R
gcd(mr , l )2
mr
≥ ∆3(l )
24
(2)
holds for 1 ≤ l ≤ L, then for positive integers n >−∆2/24, we have
g (n) = 2π ∑
l∈L>0
∆4(l )
(
∆3(l )
24n +∆2
)(∆1+1)/2 ∞∑
k=1
k≡l mod L
I∆1+1
( π
6k
√
∆3(l )(24n +∆2)
) Âk (n)
k
,
where
Iν(s) :=
∞∑
m=0
( s2 )
ν+2m
m!Γ(ν+m +1)
is the νth modified Bessel function of the first kind, and Γ(s) = ∫ ∞0 e−t t s−1 dt is the gamma
function.
3. Proofs of Theorem 1 and Corollary 2
In this section, we use Theorem 3 to prove Theorem 1. As a consequence, we obtain Corollary 2
using the asymptotic expansion of the modified Bessel function of the first kind due to Baner-
jee [2].
Proof of Theorem 1. Writing
1
(q ; q)∞(q p ; q p )∞
=
∞∑
n=0
ap (n)q
n ,
we have m = (1, p),δ= (−1,−1) and L = p. We compute ∆1 = 1 and ∆2 =−p −1. Recalling that p
is a prime, we have that for l ∈ {1, . . . , p},
∆3(l ) =
{
1+ 1p , l ̸= p
1+p, l = p , ∆4(l ) =
{p
p, l ̸= p
1, l = p ,
so that L>0 = {1, . . . , p}. Since p ≤ 23, we see that condition (2) holds for l ∈ {1, . . . , p}. Applying
Theorem 3 with Âk (n) = bk (n), we obtain
ap (n) = 2π
∑
l∈L>0
∆4(l )
(
∆3(l )
24n −p −1
) ∞∑
m=1
m≡l mod p
I2
( π
6m
√
∆3(l )(24n −p −1)
) bm(n)
m
for n > (p +1)/24. Plugging in the values of ∆3(l ) and ∆4(l ) yields the desired result. □
We see that the series for ap (n) in Theorem 1 converges rapidly and that the term m = 1
contributes signficantly to the series. We thus obtain the asymptotic formula
ap (n) ∼ 2πpp
(
1+p−1
24n −p −1
)
I2
(
π
6
√
(1+p−1)(24n −p −1)
)
, n −→∞. (3)
Russelle Guadalupe 1489
To prove Corollary 2, we use the following asymptotic formula [2, Equation (2.11)]
Iν(s) ∼ e
s
p
2πs
∞∑
m=0
(−1)mdm(ν)
xm
, s −→∞, (4)
where
dm(ν) =
(ν−1/2
m
)
(ν+ 12 )m
2m
with (
a
m
)
=
{
a(a−1)···(a−m+1)
m! , m ∈N
1, m = 0 ,
(a)m =
{
a(a +1) · · · (a +m −1), m ∈N
1, m = 0
for a ∈R.
Proof of Corollary 2. From (4) we get
I2(s) = e
s
p
2πs
(
1− 15
8s
+O
(
1
s2
))
.
In view of (3), we have
ap (n)∼2
√
3p · (1+p
−1)3/4
(24n−p−1)5/4 ·exp
(
π
6
√
(1+p−1)(24n−p−1)
)(
1− 45
8π
√
6n(1+p−1)
+O
(
1
n
))
. (5)
Since
1
(24n −p −1)5/4 =
1
(24n)5/4
(
1+O
(
1
n
))
,
exp
(
π
6
√
(1+p−1)(24n −p −1)
)
= exp
π
√
2n(1+p−1)
3
(1− p−1/2(1+p)3/2π
24
p
6n
+O
(
1
n
))
,
we infer from (5) that
ap (n) ∼ 2
√
3p · (1+p
−1)3/4
(24n)5/4
(
1+O
(
1
n
))
·exp
π
√
2n(1+p−1)
3

·
(
1− p
−1/2(1+p)3/2π
24
p
6n
+O
(
1
n
))(
1− 45
8π
√
6n(1+p−1)
+O
(
1
n
))
,
and taking logarithms yields the desired conclusion. □
We remark that when p > 23 is a prime, condition (2) of Theorem 3 fails. Nonetheless, by
numerical experiments via Mathematica, we propose the following conjecture.
Conjecture 4. Corollary 2 also holds for primes p > 23.
Acknowledgments
The author would like to thank the anonymous referee for helpful comments that improved the
contents of the paper.
1490 Russelle Guadalupe
Declaration of Interest
The author do not work for, consult, own shares in or receive funding from any company or
organization that would benefit from this article, and have declared no affiliation other than their
research organisations.
References
[1] Z. Ahmed, N. D. Baruah and M. G. Dastidar, “New congruences modulo 5 for the number
of 2-color partitions”, J. Number Theory 157 (2015), pp. 184–198.
[2] K. Banerjee, “Inequalities for the modified Bessel function of first kind of non-negative
order”, J. Math. Anal. Appl. 524 (2023), no. 1, article no. 127082 (28 pages).
[3] K. Banerjee, P. Paule, C.-S. Radu and W. Zeng, “New inequalities for p(n) and log p(n)”,
Ramanujan J. 61 (2023), no. 4, pp. 1295–1338.
[4] K. Bringmann and K. Ono, “Coefficients of harmonic Maass forms”, in Partitions, q-series,
and modular forms, Springer, 2012, pp. 23–38.
[5] O.-Y. Chan, “Some asymptotics for cranks”, Acta Arith. 120 (2005), no. 2, pp. 107–143.
[6] W. Y. C. Chen, L. X. W. Wang and G. Y. B. Xie, “Finite differences of the logarithm of the
partition function”, Math. Comput. 85 (2016), no. 298, pp. 825–847.
[7] S. Chern, “Asymptotics for the Fourier coefficients of eta-quotients”, J. Number Theory 199
(2019), pp. 168–191.
[8] S. DeSalvo and I. Pak, “Log-concavity of the partition function”, Ramanujan J. 38 (2015),
no. 1, pp. 61–73.
[9] G. H. Hardy and S. Ramanujan, “Asymptotic formulae in combinatory analysis.”, Proc.
Lond. Math. Soc. 17 (1918), pp. 75–115.
[10] B. Kim, “A crank analog on a certain kind of partition function arising from the cubic
continued fraction”, 2008. preprint, pp. 13.
[11] V. Kotesovec, “A method of finding the asymptotics of q-series based on the convolution of
generating functions”, 2015. https://arxiv.org/abs/1509.08708.
[12] L. Mauth, “Exact formula for cubic partitions”, Ramanujan J. 64 (2024), no. 4, pp. 1323–
1334.
[13] K. Ono, The web of modularity: arithmetic of the coefficients of modular forms and q-
series, Published for the Conference Board of the Mathematical Sciences, by the American
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254.
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A note on the exact formulas for certain $2$-color partitions

Mark L. Lewis

Quanfu Yan

Comptes Rendus
Mathématique
Mark L. Lewis and Quanfu Yan
On the group pseudo-algebra of finite groups
Volume 362 (2024), p. 1661-1665
Online since: 25 November 2024
https://doi.org/10.5802/crmath.671
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. 1661-1665
https://doi.org/10.5802/crmath.671
Research article / Article de recherche
Group theory, Representation theory / Théorie des groupes, Théorie des
représentations
On the group pseudo-algebra of finite groups
Sur la pseudo-algèbre de groupe des groupes finis
Mark L. Lewis ∗,a and Quanfu Yan a,b
a Department of Mathematical Sciences, Kent State University, Kent, OH 44242, USA
b School of Mathematical Sciences, Peking University, Beijing, 100871, People’s Republic of China
E-mails: lewis@math.kent.edu, qyan5@kent.edu
Abstract. Let G be a finite group. The group pseudo-algebra of G is defined as the multi-set C (G) =
{(d ,mG (d)) | d ∈ Cod(G)}, where mG (d) is the number of irreducible characters of G with codegree d ∈
Cod(G). We show that there exist two finite p-groups with distinct orders that have the same group pseudo-
algebra, providing an answer to Question 3.2 in [7]. In addition, we also discuss under what hypothesis two
p-groups with the same group pseudo-algebra will be isomorphic.
Résumé. Soit G un groupe fini. La pseudo-algèbre de groupe de G est définie comme le multi-ensemble
C (G) = {(d ,mG (d)) | d ∈ Cod(G)}, où mG (d) est le nombre de caractères irréductibles de G de codegré
d ∈ Cod(G). Nous montrons qu’il existe deux p-groupes finis avec des ordres distincts qui ont la même
pseudo-algèbre de groupe, ce qui fournit une réponse à la question 3.2 de [7]. De plus, nous discutons
également sous quelles hypothèses deux p-groupes ayant la même pseudo-algèbre de groupe sont forcément
isomorphes.
Keywords. Finite p-groups, Characters, Group pseudo-algebra.
Mots-clés. p-groupes finis, Caractères, Pseudo-algèbre de groupe.
2020 Mathematics Subject Classification. 20C15, 20D15.
Manuscript received 11 May 2024, revised 16 July 2024, accepted 26 July 2024.
1. Introduction
All groups considered in this article are finite. As usual, G will always be a finite group, and
k(G) denotes the number of conjugacy classes of G . We write Irr(G) to denote the set of complex
irreducible characters of G and cd(G) = {χ(1) | χ ∈ Irr(G)}. Let χ ∈ Irr(G). The codegree of χ is
defined as
codχ= |G : kerχ|
χ(1)
,
which was introduced by Qian, Wang and Wei in [9]. The concept has been studied extensively
and proved to have interesting connections with some algebraic structure of finite groups (see,
for example, [4, 6–8, 10]).
∗Corresponding author
ISSN (electronic): 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1662 Mark L. Lewis and Quanfu Yan
In [7], A. Moretó first introduced the concept of the group pseudo-algebra, which is defined as
the multi-set
C (G) = {(d ,mG (d)) | d ∈ Cod(G)},
where Cod(G) = {codχ | χ ∈ Irr(G)} and mG (d) is the number of irreducible characters having
codegree d . He showed that if two finite abelian groups have the same group pseudo-algebra,
then they are isomorphic. Additionally, a natural question arises: must groups have the same
order if they have the same group pseudo-algebra? A particular case of the question asks whether
G ∼= A provided that C (G) =C (A), where G is a finite group and A is an abelian p-group for some
prime p. He gave an affirmative answer when either A is cyclic or the exponent of A does not
exceed p2 (see [7, Theorem 3.4]). Thus, the next natural case to look is when the exponent of A is
p3. We will prove that if G is a nonabelian group and A is an abelian p-group of exponent p3 so
that C (G) =C (A), then p = 2. This result guides us in constructing examples, suggesting that the
question does not always yield a positive answer.
Theorem 1. Let p be a prime. There exists an abelian p-group A and a group G with C (G) =C (A)
so that G is not isomorphic to A. Hence, groups may have different orders even though they have
the same group pseudo-algebra.
From the above theorem, it is clear that the question does not always have a positive answer
when the abelian p-group A has three generators. Thus, it might be a good idea to focus on the
case when A has two generators, in particular, when A ∼= Cpn ×Cp , where p is a prime. We first
consider metacyclic groups G satisfying C (G) =C (A) and it turns out that G must be isomorphic
to A. Applying this result, we can prove that for a group G and A ∼=Cpn ×Cp , if C (G) =C (A), then
either G ∼= A or |G : G ′| = p2 and p > 2.
Theorem 2. Let G be a group and A ∼=Cpn ×Cp , where p is a prime and n ⩾ 3 is an integer. Suppose
that C (G) =C (A). Then G is a p-group and one of the following holds:
(1) G ∼= A.
(2) |G : G ′| = p2, p > 2 and Z (G) is noncyclic. In addition, there is a unique maximal subgroup
X of G ′ which is normal in G so that the factor group G/X is nonabelian of order p3 and of
exponent p.
Applying the above result, we show that G ∼= A if A ∼= Cp3 ×Cp and C (G) = C (A). We also
demonstrate that under the same hypothesis as stated in the above theorem, if, in addition, G
has either a metacyclic maximal subgroup or a two-generator derived subgroup G ′, then G ∼= A.
Theorem 3. Let G be a group and A ∼= Cpn ×Cp , where p is a prime. Suppose that C (G) = C (A).
Then G ∼= A if one of the following holds:
(1) G has a metacyclic maximal subgroup,
(2) The derived subgroup G ′ is generated by two elements,
(3) The derived subgroup G ′ is abelian.
2. Main Results
In this section, we start by stating a fact that will be used frequently. Theorem A in [9] yields that
if G is a group such that Cod(G) is a set of powers of a prime p, then G is a p-group. Now we prove
the following basic lemmas.
Lemma 4. Let G be a nonabelian group and A be an abelian p-group of order pa for some prime
p. Suppose that C (G) =C (A). Then |cd(G)|⩾ 3.
Mark L. Lewis and Quanfu Yan 1663
Proof. Since A is abelian, we have that A ∼= Irr(A) and so Cod(A) coincides with the set of element
orders of A. Hence Cod(G) = Cod(A) is a set of powers of p. It follows that G is a p-group.
Assume that |cd(G)| < 3. Then |cd(G)| = 2 as G is nonabelian. So we may assume cd(G) = {1, pe }
for some positive integer e. Let |A| = pa , |G| = pn and |G : G ′| = pr . By C (G) = C (A), we have
that k(G) = k(A) = pa . Then pr < pa < pn . Notice that |G| = |G : G ′| + (k(G)− |G : G ′|)p2e . Thus
pn−r −1 = (pa−r −1)p2e , contrary to the fact that p2e does not divide pn−r −1. Hence |cd(G)|⩾ 3,
as wanted. □
Now we delve deeper into the degree set of G . In particular, we consider the case when
cd(G) = {1, p, p2}.
Lemma 5. Let G be a nonabelian group and A be an abelian p-group of order pa for some prime
p. Suppose that C (G) = C (A) and cd(G) = {1, p, p2}. Then p = 2 and |G| = 2a+2. In particular, if
we write |G : G ′| = pr and k1 = |{χ ∈ Irr(G) | χ(1) = p}| and k2 = |{χ ∈ Irr(G) | χ(1) = p2}|, then
k1 = pa −pr −pr−2 and k2 = pr−2.
Proof. Following the same reasoning process as in Lemma 4, G is a p-group. Since C (G) =C (A),
we have that k(G) = k(A) = pa and so |G| > pa as G is nonabelian. On the other hand, we have
that
|G| = |G : G ′|+k1p2 +k2p4
⩽ pr +p2 + (pa −pr −1)p4
= pa+4 −pr+4 +pr −p4 +p2
< pa+4.
Hence, |G| = pa+1, pa+2, or pa+3. Notice that |G : G ′| = k(G)−k1 −k2 and so |G| = k(G)+k1(p2 −
1)+k2(p4 − 1). Hence (p2 − 1) | (|G| −k(G)), which indicates that |G| = pa+2. By [5, Theorem 3],
such groups do not exist if p is odd. Hence, p = 2. Now it is easy to see that k1 = pa −pr −pr−2
and k2 = pr−2. □
With the above lemmas, we are prepared to provide an example for Theorem 1, and it is
advisable to look at 2-groups. Let A ∼= C23 × C2 × C2 be an abelian 2-group. Then C (A) =
{(1,1), (2,7), (22,8), (23,16)}. Since χ(1) < codχ for all non-principal characters χ of G , it fol-
lows that cd(G) is a subset of {1,2,22}. By Lemma 4 and Lemma 5, if there is a group G so
that C (G) = C (A), then |G| = 27 and cd(G) = {1,2,4}. We notice that such a group does ex-
ist. For example, using GAP, G can be one of SmallGroup(128,755), SmallGroup(128,756),
SmallGroup(128,773). To enhance the readability, we present the information of the irre-
ducible representations of SmallGroup(128,773) (see https://people.maths.bris.ac.uk/~matyd/
GroupNames/128/C4sC4s7D4.html)
Since there is a counterexample when A has three generators, we move on to the case when
A =Cpn ×Cp , where p is a prime. We first give the following lemma.
Lemma 6. Let G be a group and A = Cpn ×Cp , where p is a prime. Suppose that C (G) = C (A).
Then G is a p-group and G/G ′ ∼=Cpm ×Cp for some integer m ⩽ n.
1664 Mark L. Lewis and Quanfu Yan
Proof. It is clear that [9, Theorem A] implies that G is a p-group. By [7, Lemma 3.3], G/Φ(G) ∼=
A/Φ(A) ∼=Cp ×Cp . Notice that G ′ ⩽Φ(G) andΦ(G)/G ′ ∼=Φ(G/G ′). Then G/G ′ has two generators.
It follows that G/G ′ ∼= Cpm ×Cp l for some positive integers m and l . If both m and l are greater
than 1, then it is easy to see that G has more irreducible characters of codegree p2 than the abelian
group A. This is impossible. Hence, without loss of generality, we can let l = 1. On the other hand,
it is clear that m ⩽ n. The proof is complete now. □
We now consider a class of two-generator groups: metacyclic groups. We obtain the following
result.
Theorem 7. Let G be a metacyclic group and A = Cpn ×Cp , where p is a prime. Suppose that
C (G) =C (A). Then G is abelian and G ∼= A.
Proof. We write G = HK , where H and K are cyclic subgroups of G and H is normal in G . If either
one of these has index p, then by ([1, Theorem 1.2]), k(G) can be computed and it is not a power
of p. Therefore, both H and K have index greater than or equal to p2. Then the group G maps
onto a group, say M , of order p4, which is a product of two cyclic groups of order p2. Clearly, the
irreducible characters of M can be viewed as irreducible characters of G . If M is abelian, then
M ∼= Cp2 ×Cp2 . Notice that C (G) = C (A) = {(1,1), (p, p2 − 1), (p2, p2(p − 1)), . . . , (pn , pn(p − 1))}.
Hence, M has more irreducible characters of codegree p2 than G . Thus M is nonabelian. It
follows that Z (M) is noncyclic of order p2 and so cd(M) = {1, p} and |kerχ| = p for all nonlinear
characters χ ∈ Irr(M). In other words, all nonlinear characters of M have codegree p2. Notice that
M ′ ⩽ Z (M) and M ′ is cyclic. So M ′ has order p. Since M/M ′ must be isomorphic to a subgroup of
A, we have that M/M ′ ∼=Cp2×Cp . Hence, there are p2(p−1) linear characters having codegree p2.
Together with those nonlinear characters, there are more than p2(p −1) irreducible characters of
codegree p2. □
Next, we give a proof of Theorem 2.
Proof of Theorem 2. If G is abelian, then (1) follows. Assume now that G is nonabelian. By
Lemma 6, we can write G/G ′ ∼=Cpa−1 ×Cp , where 2 ⩽ a ⩽ n. A proof similar to Theorem 7 shows
that G is a p-group, k(G) = k(A) = pn+1 and |G|⩾ pn+3. Since G is a nonabelian p-group with two
generators, by [3, Lemma 2.2] we can let X be the unique maximal subgroup of G ′ which is normal
in G . Consider the factor group G =G/X . Then |G| = pa+1, G/G ′ ∼=G/G ′, and G ′ =G ′/X has order
p. Notice that for any irreducible character χ ∈ Irr(G) with χ(1) > 1, codχ = pa+1χ(1)|kerχ| ⩾ pa . Then
χ(1)⩽ p and kerχ= 1. This implies that cd(G) = {1, p} and all nonlinear irreducible characters are
faithful. Hence G ′ is the unique minimal normal subgroup of G .
Write G/G ′ = C /G ′ ×D/G ′, where C /G ′ ∼= Cpa−1 and D/G ′ ∼= Cp . Next, we will discuss in two
cases.
Case 1: |G : G ′| > p2. We claim that that G is metacyclic. By [3, Theorem 2.3], we only need to
show that G is metacyclic. If Φ(C ) = 1, then C is elementary abelian and so C /G ′ ∼=Cp and a = 2,
contrary to |G : G ′| > p2. As Φ(C ) char C ⩽G , it follows that Φ(C ) ⩽G . Hence by the uniqueness
of G ′, we have that G ′ ⩽Φ(C ). Notice that (C /G ′)/(Φ(C )/G ′) ∼= C /Φ(C ) is cyclic. Then C is cyclic.
Since G/C is cyclic, G is metacyclic. Hence, the above claim holds. It follows from Theorem 7 that
G is abelian, which is a contradiction. Hence, this case cannot happen.
Case 2: |G : G ′| = p2. If p = 2, then |G : G ′| = 4 and such groups have been classified. By the
equation |G| = |G : G ′|+ (k(G)−|G : G ′|) ·22, it is easy to see that k(G) is not a power of 2, contrary
to the hypothesis C (G) = C (A). Hence p > 2. Since G/G ′ ∼= G/G ′ ∼= Cp ×Cp , it follows that G has
order p3. Now we only need to show that G has exponent p. If G has exponent p2, then the group
C defined above is cyclic of order p2 and hence G is metacyclic and so G must be metacyclic. It
Mark L. Lewis and Quanfu Yan 1665
follows from Theorem 7 that G is abelian, which is a contradiction. Hence G is of exponent p and
so (2) follows. □
By Lemma 4 and Lemma 5, we have that if A ∼=Cp3×Cp and G is a group satisfying C (G) =C (A),
then G ∼= A. Theorem 3 immediately follows from Theorem 2 and a result of Blackburn. He proved
that if a p-group G and its derived subgroup G ′ are generated by two elements, then G ′ is abelian
(see [2, Theorem 4]).
Proof of Theorem 3. If G is abelian, there is nothing to prove. If G is not isomorphic to A, then
by Theorem 2, we have that |G : G ′| = p2 and p > 2. If G has a metacyclic maximal subgroup M ,
then G ′ ⩽Φ(G) is a subgroup of M and so G ′ is metacyclic, which indicates that G ′ is generated
by two elements. Hence by Blackburn’s result, G ′ is abelian. Now we have that G ′ is abelian in all
three cases. Notice that χ(1) | |G : G ′| for all irreducible characters χ of G . Then cd(G) is a subset
of {1, p, p2}. It follows from Lemma 4 and 5 that p = 2. This is a contradiction. Therefore, G ∼= A,
as desired. □
Acknowledgements
The work in this paper was completed by the second author (Ph.D. student) under the supervision
of the first author at Kent State University. The contents of this paper appear as part of the second
author’s Ph.D. dissertation. We would like to thank Professor Steve Gagola for several helpful
discussions while writing this paper. In particular, the proof of Theorem 7 presented here is
largely due to an idea of his which is much simpler than our original proof. We would like to
thank the referee for a careful reading of the paper.
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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