Comptes Rendus Mathématique

Duc Hiep Pham

Comptes Rendus
Mathématique
Duc Hiep Pham
p-adic non-commutative analytic subgroup theorem
Volume 360 (2022), p. 933-936
https://doi.org/10.5802/crmath.325
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 933-936
https://doi.org/10.5802/crmath.325
Number theory / Théorie des nombres
p-adic non-commutative analytic subgroup
theorem
Duc Hiep Phama
a University of Education, Vietnam National University, Hanoi, 144 Xuan Thuy, Cau
Giay, Hanoi, Vietnam
E-mail: phamduchiep@vnu.edu.vn
In memory of my father, Duc Hoa Pham
Abstract. In this paper, we formulate and prove the so-called p-adic non-commutative analytic subgroup
theorem. This result is seen as the p-adic analogue of a recent theorem given by Yafaev in [11].
2020 Mathematics Subject Classification. 14L10, 22E35, 11F85, 11J81.
Manuscript received 15 July 2021, revised 20 September 2021, accepted 10 January 2022.
1. Introduction
It is known that the analytic subgroup theorem is considered as one of the most powerful
theorems in complex transcendental number theory. The theorem was established by Wüstholz
in the 1980’s based on a very deep auxiliary result on multiplicity estimates on group varieties
(see [8] and [9]). To present the theorem, we start with G a commutative algebraic group defined
over Q, that is a smooth quasi-projective variety defined over Q with a commutative group law
for which the composition map G ×G → G , (g ,h) 7→ g ◦h and the inverse map G → G , g 7→ g−1
are regular morphisms between algebraic varieties defined over Q. Then the set G(C) of complex
points of G is a complex Lie group, and one has the exponential map expG(C) : Lie(G(C)) → G(C)
of G . We say that an element u ∈ Lie(G(C)) is an algebraic point of the exponential map of G if
expG(C)(u) ∈G(Q). The following theorem is a direct consequence of [7, Theorem 1], and it is still
called the analytic subgroup theorem (see also [6, Theorem 1.2]).
Theorem 1 (Wüstholz). Let G be a connected commutative algebraic group defined over Q. Let
u ∈ Lie(G(C)) be an algebraic point of the exponential map of G and Vu the smallest Q-vector
subspace of Lie(G) such that u lies in the complex vector space Vu⊗QC. Then there exists a connected
algebraic subgroup Hu of G defined overQ satisfying Lie(Hu) =Vu .
Many results in transcendence theory can be deduced from Wüstholz analytic subgroup
theorem as consequences by choosing suitable commutative algebraic groups. For instance,
the theorem generalizes Baker’s famous result on linear forms in logarithms and its elliptic
analogue in full generality (see [1, Chapter 6]). It implies some important results concerning
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
934 Duc Hiep Pham
linear independence of periods of abelian varieties (see [6, Chapter 1]). Furthermore, Wüstholz
himself has recently obtained a nice application on elliptic and abelian periods spaces (see[10]).
It is possible to extend Theorem 1 to the non-commutative case. Note that in this case, there
is still the exponential map between the Lie algebra and the Lie group as the commutative case.
The following theorem has been recently proved by Yafaev (see [11]).
Theorem 2 (Yafaev). Let G be a connected algebraic group defined overQ. Let u ∈ Lie(G(C)) be an
algebraic point of the exponential map of G and Vu the smallest Q-vector subspace of Lie(G) such
that u lies in the complex vector space Vu ⊗Q C. Then there exists a connected algebraic subgroup
Hu of G defined overQwhich is commutative and satisfies Lie(Hu) =Vu .
In 2015, Fuch and Pham obtained the p-adic analytic subgroup theorem (see [3]), and the aim
of this paper is to obtain a p-adic analogue of Theorem 2 above. In the p-adic setting, similar to [3,
Theorem 2.2], we also use the p-adic logarithm map which is still valid in the non-commutative
case. Fix a prime number p, let Qp be the field of p-adic numbers. Denote by Cp the completion
of the algebraic closure of Qp (with respect to the normalized p-adic absolute value). Let G now
be an algebraic group defined over Q. Consider the set G(Cp ) of Cp -points of G . This is a Lie
group over Cp . Denote by G(Cp ) f the set of x in G(Cp ) satisfying the condition that there exists a
strictly increasing sequence (ni ) of positive integers for which xni tends to the identity element
of G(Cp ) as i tends to infinity. It follows from [2, Chapter III, 7.6] that there is a Cp -analytic map
logG(Cp ) : G(Cp ) f → Lie(G(Cp )), and it is called the p-adic logarithm map of G . Then the p-adic
non-commutative analytic subgroup theorem is formulated as follows.
Theorem 3. Let G be a connected algebraic group defined over Q of positive dimension, and
logG(Cp ) : G(Cp ) f → Lie(G(Cp )) the p-adic logarithm map of G. Let γ ∈ G(Cp ) f be an algebraic
point of G(Q) with logG(Cp )(γ) 6= 0 and Vγ the smallest Q-vector subspace of Lie(G) such that
the p-adic vector space Vγ ⊗Q Cp contains logG(Cp )(γ). Then there exists a connected commuta-
tive algebraic subgroup Hγ ⊆ G defined over Q of positive dimension satisfying γ ∈ Hγ(Q) and
Lie(Hγ) =V .
The proof of our theorem follows closely that of Yafaev’s theorem. We also reduce the general
case to the commutative case and then apply the p-adic analytic subgroup theorem. The main
difference is that, in the p-adic setting, the p-adic exponential map is only defined locally on an
open subgroup of the p-adic Lie algebra of the given algebraic group. However, by using the p-
adic logarithm map and by taking a certain power of the algebraic point, we are able to still keep
the important similar arguments as in the complex case, and based on this to get the derised
result.
2. The p-adic exponential map and related commutative algebraic subgroups
In this section, we first briefly recall some background on exponential and logarithm maps over
p-adic fields. The main reference we follow here is [2]. For an algebraic group G defined over
Q, by [2, Chapter III, 7.2, 7.6] there exist an open subgroup Up of Lie(G(Cp )) and an analytic map
expG(Cp ) : Up →G(Cp ) such that expG(Cp )(Up ) is an open subgroup of G(Cp ), and the map expG(Cp )
induces an isomorphism between Up and expG(Cp )(Up ) ⊆G(Cp ) f , whose inverse is the restriction
of logG(Cp ) to expG(Cp )(Gp ). The map expG(Cp ) is called the p-adic exponential map of G . For an
element u in Up and for a subset S in Cp , define S ·u by the set {su; s ∈ S}. The following lemma
plays a central role in the proof of the main theorem which allows us to construct a relevant
commutative algebraic subgroup of G defined overQ.
Duc Hiep Pham 935
Lemma 4. Let G be an algebraic group defined overQ. Let u be an element in Up with expG(Cp )(u) ∈
G(Q). Denote by Gu the Zariski closure of the set expG(Cp )((Cp ·u)∩Up ) in G(Cp ). Then Gu is a
commutative algebraic subgroup of G defined overQ and expG(Cp )(u) ∈Gu(Q).
Proof. PutΦu := expG(Cp )((Cp ·u)∩Up ). We first see that any two elements x, y ∈Cp ·u commute,
and therefore
expG(Cp )(x)expG(Cp )(y) = expG(Cp )(x + y) = expG(Cp )(y +x) = expG(Cp )(y)expG(Cp )(x),
by Campbell–Hausdorff formula (see [5, Section 16]). In particular, this shows that Φu is a
commutative subgroup of G(Cp ). It follows from [4, Lemma 1.40] that the Zariski closure Gu ofΦu
is an algebraic subgroup of G(Cp ). We now consider the commutator morphism f : Gu ×Gu →Gu
given by f (x, y) = [x, y]. Then f is an algebraic morphism and trivial onΦu×Φu . This givesΦu×Φu
is contained in the algebraic set f −1(e), where e denotes the identity element of G . But Φu ×Φu
is Zariski-dense in Gu ×Gu , this implies that the morphism f must be trivial on Gu ×Gu . In other
words, Gu is commutative. It remains to show that Gu is defined over Q. If u = 0 then Gu is the
trivial group, and hence clearly defined over Q. Assume u 6= 0, let Γu denote the image of the
set (Q ·u)∩Up under the map expG(Cp ). As above, we also have that Γu is a subgroup of G(Cp ),
and therefore by [4, Lemma 1.40] again the Zariski closure G ′u of Γu in G(Cp ) is an algebraic
group. On the other hand, since expG(Cp )(u) ∈G(Q), it follows that expG(Cp )(u) ∈Gu(Q). Note that
expG(Cp )(mu) = expG(Cp )(u)m for all m ∈Z, and this leads to Γu is contained in Gu(Q). Hence, the
algebraic group G ′u must be defined over Q. Moreover, the group Γu is infinite (since u 6= 0), thus
the algebraic group G ′u has positive dimension.
Next, one has expG(Cp )(u) ∈G ′u(Q)∩G(Cp ) f ⊆G ′u(Cp ) f . This gives
u = logG(Cp )
(
expG(Cp )(u)
)= logG ′u (Cp ) (expG(Cp )(u)) ∈ Lie(G ′u(Cp )).
Since Lie(G ′u(Cp )) is a vector space over Cp , it follows that the line Cp · u is also contained in
Lie(G ′u(Cp )). From this, we get
Φu ⊆ expG(Cp )
(
Lie(G ′u(Cp ))∩Up
)= expG ′u (Cp ) (Lie(G ′u(Cp ))∩Up)⊆G ′u(Cp ).
This allows us to deduce that Gu(Cp ) ⊆G ′u(Cp ). But, by definition, G ′u(Cp ) is obviously contained
in Gu(Cp ). Therefore, we conclude that Gu =G ′u , and this completes the proof of the lemma. 
3. Proof of the main theorem
We are now ready to prove the main theorem. Let expG(Cp ) : Up →G(Cp ) be the p-adic exponential
map of G defined as in Section 2. By definition of the set G(Cp ) f , there is a positive integer k such
that γk ∈Up . Let u be the point logG(Cp )(γk ). We have
expG(Cp )(u) = expG(Cp )
(
logG(Cp )(γ
k )
)= γk ∈G(Q).
Using Lemma 4, there exists a commutative algebraic subgroup Gu of G defined overQ of positive
dimension with expG(Cp )(u) ∈Gu(Q). This also gives expG(Cp )(u) ∈Gu(Cp ) f , and hence
logG(Cp )(γ
k ) = u ∈ Lie(Gu(Cp )) ⊆ Lie(G(Cp )).
On the other hand, it follows from [2, Chapter III, 7.6] that
logG(Cp )(γ
k ) = k logG(Cp )(γ) ∈Vγ⊗QCp .
In particular, this implies that Vγ ⊆ Lie(Gu). This enables us to apply the p-adic analytic subgroup
theorem ([3, Theorem 2.2]) to the commutative algebraic group Gu , the algebraic point γk and
the Q-vector space Vγ to obtain an algebraic (commutative) subgroup H of G defined over Q of
positive dimension such that Lie(H) ⊆Vγ and γk ∈ H(Q). Let H 0 be the connected component of
936 Duc Hiep Pham
H , then Lie(H 0) = Lie(H) and there is a positive integer m for which (γk )m ∈ H 0(Q). Put n = km
and take the algebraic subgroup Hγ of Gu with Hγ(Q) = {α ∈ Gu(Q) : αn ∈ H 0(Q)}. Then γ ∈ Hγ
and Lie(Hγ) ⊆Vγ. This leads to
logG(Cp )(γ) = logHγ(Cp )(γ) ∈ Lie(Hγ(Cp )) = Lie(Hγ)⊗QCp .
By the minimality of the Q-vector space Vγ, we conclude that Lie(Hγ) = Vγ. The theorem is
therefore proved. 
Remark. Using the same arguments as in the proof of [11, Theorem 1.5] to deduce it from
the non-commutative analytic subgroup theorem, we use our p-adic analogue to deduce the
following:
Theorem 5. Let G be an algebraic group defined over Q and expG(Cp ) : Up →G(Cp ) be the p-adic
exponential map of G. Let u be a non-zero element of Up∩Lie(G) such that expG(Cp )(u) lies in G(Q).
(1) The element expG(Cp )(u) is contained in a subgroup isomorphic to the additive group Ga .
(2) Assume G is an affine algebraic group. The element u is a nilpotent element of Lie(G(Cp )),
and hence expG(Cp )(u) is a unipotent element of G(Cp ).
Acknowledgements
The author would like to thank the anonymous referee for careful reading of this manuscript and
useful comments.
References
[1] A. Baker, G. Wüstholz, Logarithmic Forms and Diophantine Geometry, New Mathematical Monographs, vol. 9,
Cambridge University Press, 2007.
[2] N. Bourbaki, Elements of Mathematics. Lie groups and Lie algebras. Part I: Chapters 1-3. English translantion,
Actualités Scientifiques et Industrielles, Addison-Wesley Publishing Group, 1975.
[3] C. Fuchs, D. H. Pham, “The p-adic analytic subgroup theorem revisited”, p-Adic Numbers Ultrametric Anal. Appl. 7
(2015), no. 2, p. 143-156.
[4] J. S. Milne, Algebraic Groups. The theory of group schemes of finite type over a field, Cambridge Studies in Advanced
Mathematics, vol. 170, Cambridge University Press, 2017.
[5] P. Schneider, p-adic Lie groups, Grundlehren der Mathematischen Wissenschaften, Springer, 2011.
[6] P. Tretkoff, Periods and Special Functions in Transcendence, Advanced Textbooks in Mathematics, World Scientific,
2017.
[7] G. Wüstholz, “Some remarks on a conjecture of Waldschmidt”, in Approximations diophantiennes et nombres
transcendants, Progress in Mathematics, vol. 31, Birkhäuser, 1983, p. 329-336.
[8] ——— , “Multiplicity estimates on group varieties”, Ann. Math. 129 (1989), p. 471-500.
[9] ——— , “Algebraische Punkte auf Analytischen Untergruppen algebraischer Gruppen”, Ann. Math. 129 (1989),
p. 501-517.
[10] ——— , “Elliptic and abelian period spaces”, Acta Arith. 198 (2021), p. 329-357.
[11] A. Yafaev, “Non-commutative analytic subgroup theorem”, J. Number Theory 230 (2022), p. 233-237.


$p$-adic non-commutative analytic subgroup theorem

Hamdi Zorgati

A $\Gamma $-convergence result for optimal design problems

Bamdad Hosseini

Stefan Steinerberger

Comptes Rendus
Mathématique
Bamdad Hosseini and Stefan Steinerberger
Intrinsic Sparsity of Kantorovich solutions
Volume 360 (2022), p. 1173-1175
https://doi.org/10.5802/crmath.392
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
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e-ISSN : 1778-3569
Comptes Rendus
Mathématique
2022, Vol. 360, p. 1173-1175
https://doi.org/10.5802/crmath.392
Control theory / Théorie du contrôle
Intrinsic Sparsity of Kantorovich solutions
Parcité intrinsèque des solutions de Kantorovich
Bamdad Hosseinia and Stefan Steinerberger∗, b
a Department of Applied Mathematics, University of Washington, Seattle, USA
b Department of Mathematics, University of Washington, Seattle, USA
E-mails: bamdadh@uw.edu (BH), steinerb@uw.edu (SS)
Abstract. Let X ,Y be two finite sets of points having #X = m and #Y = n points withµ= (1/m)(δx1 +·· ·+δxm )
and ν= (1/n)(δy1 +·· ·+δyn ) being the associated uniform probability measures. A result of Birkhoff implies
that if m = n, then the Kantorovich problem has a solution which also solves the Monge problem: optimal
transport can be realized with a bijection π : X → Y . This is impossible when m 6= n. We observe that when
m 6= n, there exists a solution of the Kantorovich problem such that the mass of each point in X is moved to
at most n/gcd(m,n) different points in Y and that, conversely, each point in Y receives mass from at most
m/gcd(m,n) points in X .
Résumé. Soient X et Y deux ensembles de points de cardinaux respectifs m et n; et µ,ν les mesures de
probabilité associées. Quand m = n, un résultat de Birkhoff assure que le problème de Kantorovitch a une
solution qui résout aussi le problème de Monge : une bijection réalise le transport optimal. Quand m 6= n,
nous montrons que le problème de Kantorovitch admet une solution telle que chaque point de X se déplace
vers au plus n/pgcd(m,n) points de Y, et réciproquement, chaque point de Y reçoit de la masse d’au plus
m/pgcd(m,n) points de X.
2020 Mathematics Subject Classification. 49Q20, 90C46.
Funding. Both authors gratefully acknowledge support from the Kantorovich Initiative. SS is partially sup-
ported by the NSF (DMS-2123224) and the Alfred P. Sloan Foundation.
Manuscript received 3 June 2022, accepted 5 June 2022.
1. Introduction
Let µ and ν be two (probability) measures. A classical question, due to Monge, is to understand
the optimal way of mapping µ to ν. If we denote the cost of transporting mass from x to y by
c(x, y), then the Monge problem asks for
inf
T
{∫
X
c (x,T (x))dµ(x) : T∗(µ) = ν
}
(Monge),
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
1174 Bamdad Hosseini and Stefan Steinerberger
where T∗(µ) denotes the push forward of µ by T . This problem may not be solvable because such
transport maps T may simply not exist. Kantorovich proposed to relax the problem and instead
try to minimize
inf
γ
∫
X×Y
c(x, y)dγ(x, y) (Kantorovich),
where γ is a probability measure on X ×Y having marginalsµ and ν. There is a nice classical result
linking these two problems in the discrete setting: ifµ andν are two uniform probability measures
over two sets X and Y with n elements, then it is known that these two problems coincide.
Theorem 1 (see e.g. [2, 4, 6]). If µ= (1/n)∑ni=1δxi and ν= (1/n)∑ni=1δyi , then there is a solution
of the Kantorovich problem which also solves the Monge problem.
The statement is independent of the transport costs c(xi , y j ). A proof is as follows: the Kan-
torovich problem is, in the discrete setting, a linear program over bistochastic matrices. A theo-
rem of Birkhoff [1] (also attributed to König [3] and von Neumann [5]) says that the bistochastic
matrices are the convex hull of the permutation matrices. The minimum of a linear program in a
non-empty polyhedron is attained in an extremal point.
2. Main Result
No such statement can be true when m 6= n: the two sets have different cardinalities and no
bijection is possible. The goal of this short note is to point out that there nonetheless exists a
particularly simple solution of the Kantorovich problem.
Theorem 2. Let µ= (1/m)∑mi=1δxi and ν= (1/n)∑ni=1δyi . There is a solution of the Kantorovich
problem such that mass from each point in X is moved to at most n/gcd(m,n) different points in
Y and that each point in Y receives mass from at most m/gcd(m,n) points in X .
Figure 1. Left: m = n, the transport is a bijection. Right: m = 20 red points are sent to n = 30
blue points. Each red point is transported to at most 30/gcd(20,30) = 3 blue points, each
blue points receives mass from at most 20/gcd(20,30) = 2 red points.
Somewhat to our surprise, we were unable to find this simple but intriguing statement (illus-
trated in Fig. 1) in the literature. Besides its intrinsic appeal, it does seem like it could be poten-
tially useful insofar as it guarantees the existence of “sparse” solutions of the Kantorovich prob-
lem (with sparsity depending on m,n).
Proof. Suppose that
µ= 1
m
m∑
i=1
δxi and ν=
1
n
n∑
i=1
δyi
Bamdad Hosseini and Stefan Steinerberger 1175
are two given measures. We replace each point xi by n/gcd(m,n) identical points xi , j for 1 ≤ j ≤
n/gcd(m,n) and, likewise, we replace each point yi by m/gcd(m,n) identical points yi , j where
1 ≤ j ≤ m/gcd(m,n). This allows us to write
µ= gcd(m,n)
mn
m∑
i=1
n/gcd(m,n)∑
j=1
δxi , j and ν=
gcd(m,n)
mn
n∑
i=1
m/gcd(m,n)∑
j=1
δyi , j .
The problem can now be interpreted as finding a transport map from mn/gcd(m,n) points
of the same weight to another set of mn/gcd(m,n) points of the same weight. Applying the
classical result shows that there exists bijective map between the points that realizes the optimal
Kantorovich cost. This corresponds into each point in X being split into at most n/gcd(m,n)
equal parts and each point in Y being split into at most m/gcd(m,n) parts. 
We conclude by observing that the same argument also applies to linear combinations of
weighted Dirac measures as long as the weights are rational. Suppose
µ=
m∑
i=1
ai
bi
δxi where
m∑
i=1
ai
bi
= 1
and where ai ,bi ∈ N are positive rational weights. We note that it is possible to equivalently
represent µ as a linear combination of a number of equally weighted Dirac measures. This
number will depend on the least common multiple lcm(b1, . . . , bm) of the denominators. This
can be seen by writing
µ= 1
lcm(b1, . . . , bm)
m∑
i=1
ai lcm(b1, . . . , bm)
bi
δxi
and noting that ai lcm(b1, . . . , bn)/bi ∈N. This implies the following corollary.
Corollary 3. Let
µ=
m∑
i=1
ai
bi
δxi and ν=
n∑
i=1
ci
di
δyi
be two probability measures with positive rational weights and let
B = lcm(b1, . . . , bm) and D = lcm(d1, . . . , dn).
There exists a solution of the Kantorovich problem such that mass from each point in X is moved
to at most D/gcd(B ,D) different points in Y and each point in Y receives mass from at most
B/gcd(B ,D) different points in X .
We note that this reduces to the previous Theorem when ai = ci = 1, bi = m and di = n. We
also observe that it is quite possible that D/gcd(B ,D) À n and that B/gcd(B ,D) À m (this will
usually happen when the bi ,di have many different prime factors). In such a case, the statement
would not say anything of interest.
References
[1] G. Birkhoff, “Tres observaciones sobre el algebra lineal”, Rev., Ser. A, Univ. Nac. Tucumán 5 (1946), p. 147-151.
[2] H. Brezis, “Remarks on the Monge-Kantorovich problem in the discrete setting”, C. R. Math. Acad. Sci. Paris 356 (2018),
no. 2, p. 207-213.
[3] D. König, Theorie der endlichen und unendlichen Graphen, Teubner-Archiv zur Mathematik, vol. 6, Teubner, 1936.
[4] Q. Mérigot, B. Thibert, “Optimal transport: discretization and algorithms”, in Geometric partial differential equations.
Part 2, Handbook of Numerical Analysis, vol. 22, Elsevier, 2021, p. 133-212.
[5] J. von Neumann, “A certain zero-sum two-person game equivalent to an optimal assignment problem”, in Contribu-
tions to the theory of games. Vol. 2, Annals of Mathematics Studies, vol. 28, Princeton University Press, 1953, p. 5-12.
[6] G. Peyré, M. Cuturi, “Computational optimal transport: With applications to data science”, Found. Trends Mach.
Learn. 11 (2018), no. 5-6, p. 355-407.


Intrinsic Sparsity of Kantorovich solutions

Pietro Corvaja

Julian L. Demeio

Andrei S. Rapinchuk

Jinbo Ren

Umberto M. Zannier

Bounded Generation by semi-simple elements: quantitative results

Bing-Ling Wu

Xiao-Hui Yan

Comptes Rendus
Mathématique
Bing-Ling Wu and Xiao-Hui Yan
On the denominators of harmonic numbers. IV
Volume 360 (2022), p. 53-57
<https://doi.org/10.5802/crmath.282>
© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
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Comptes Rendus
Mathématique
2022, 360, p. 53-57
https://doi.org/10.5802/crmath.282
Number theory / Théorie des nombres
On the denominators of harmonic numbers.
IV
Bing-LingWua and Xiao-Hui Yan∗, b
a School of Science, Nanjing University of Posts and Telecommunications, Nanjing
210023, P. R. China
b School of Mathematics and Statistics, Anhui Normal University, Wuhu 241002,
P. R. China
E-mails: wubingling911219@163.com, yanxiaohui_1992@163.com
Abstract. Let L be the set of all positive integers n such that the denominator of 1+1/2+·· ·+1/n is less than
the least common multiple of 1,2, . . . ,n. In this paper, under a certain assumption on linear independence, we
prove that the set L has the upper asymptotic density 1. The assumption follows from Schanuel’s conjecture.
Keywords. harmonic numbers, least common multiples, upper asymptotic density.
Mathematical subject classification (2010). 11B05, 11B75.
Funding. This work is supported by the National Natural Science Foundation of China (Grant Nos. 12101332
and 12101009), the Natural Science Foundation in Jiangsu Province (Grant No. BK20200748), the Natural
Science Foundation for Colleges and Universities in Jiangsu Province (Grant No. 20KJB110003), the Anhui
Provincial Natural Science Foundation (Grant No. 2108085QA02) and the Talent Foundation of Anhui Normal
University (Grant No.752038).
Manuscript received 11th August 2021, revised 25th August 2021 and 18th September 2021, accepted 12th Oc-
tober 2021.
1. Introduction
For any positive integer n, let
Hn = 1+ 1
2
+ 1
3
+·· ·+ 1
n
= un
vn
, (un , vn) = 1, vn > 0.
The number Hn is called n-th harmonic number. Shiu [6] proved that vn = vn+1 for infinitely
many positive integers n. Recently, Wu and Chen [9] showed that the set of positive integers n
with vn = vn+1 has asymptotic density one. For related research, one may refer to [1, 2, 5, 7, 8].
Especially, Eswarathasan and Levine [2] conjectured that the set Jp of positive integers n such
that p | un is finite for any prime number p. Sanna [5] proved that Jp (x) ≤ 129p2/3x0.765, where
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
54 Bing-Ling Wu and Xiao-Hui Yan
Jp (x) denotes the number of integers which are in Jp and do not exceed x. Later, Wu and Chen [7]
improved this result to Jp (x) ≤ 3x2/3+1/(25log p).
Let L be the set of all positive integers n such that vn is less than the least common multiple
of 1,2, . . . ,n. Note that vn divides the least common multiple of 1,2, . . . ,n. Let
d̄(L ) = limsup
x→∞
L (x)
x
.
In this paper, under a certain assumption on linear independence, we prove that the set L has
the upper asymptotic density 1. Firstly, we introduce a special case of Schanuel’s conjecture [4,
p. 30–31].
Weak Schanuel’s Conjecture. If β1, . . . ,βm are non-zero, multiplicatively independent algebraic
numbers, then logβ1, . . . , logβm are algebraically independent.
It is clear that the set of prime numbers is multiplicatively independent. For any distinct
primes q1, q2, . . . , ql , it follows from weak Schanuel’s conjecture that log q1, . . . , log ql are alge-
braically independent and so are 1/log q1, . . . ,1/log ql . So we mention the following conjecture.
Conjecture 1. For any distinct primes q1, q2, . . . , ql , the l real numbers 1/log q1, . . . ,1/log ql are
linear independent overQ.
In this paper, we prove the following result.
Theorem 2. Assuming Conjecture 1, we have d̄(L ) = 1.
2. Preliminaries
Lemma 3 ( [3, Theorem 429], Mertens’ theorem).∏
p≤x
p is a prime
(
1− 1
p
)
∼ e
−γ
log x
as x →+∞.
Lemma 4 ( [3, Theorem 442], Kronecker’s theorem). If ϑ1,ϑ2, . . . ,ϑk ,1 are real numbers and
linearly independent overQ,α1,α2, . . . ,αk are arbitrary real numbers, and N and ε are positive real
numbers, then there are integers n > N , s1, s2, . . . , sk such that |nϑm − sm −αm | < ε (m = 1,2, . . . ,k).
Lemma 5. Let a,b,c,d be positive real numbers. Then
|(a,b) \ (c,d)| ≤ |c −a|+ |b −d |+2,
where (x, y) denotes the set of all integers n with x < n < y.
Proof. If b ≤ c or d ≤ a, then the claim is obvious, since (a,b)∩ (c,d) =;. So assuming b > c and
d > a one gets
|(a,b) \ (c,d)| = ∑
a<n<b
n≤c or n≥d
1 ≤ ∑
a<n≤min{b,c}
1+ ∑
max{a,d}≤n<b
1 = ∑
a<n≤c
1+ ∑
d≤n<b
1
≤ |c −a|+ |b −d |+2.
This completes the proof of Lemma 5. 
C. R. Mathématique — 2022, 360, 53-57
Bing-Ling Wu and Xiao-Hui Yan 55
3. Proof of Theorem 2
Let Ln be the least common multiple of 1,2, . . . ,n. Let pi be i -th prime, a1 = 1 and
ai =
i∏
j=2
(
1− 1
p j
)
, i = 2,3, . . . .
By Lemma 3, ai → 0 as i →+∞. For any 0 < ε < 1, we can choose k (fixed) such that ak < 12ε. In
view of the assumption,
log p2
log pi
, i = 2,3, . . . ,k
are linear independent overQ. By Lemma 4, for any
0 < δ< ε
16k log pk
, (1)
there are infinitely many k tuples q > 0, s2, . . . , sk of integers such that∣∣∣∣q log p2log pi − si + log ai−1log pi
∣∣∣∣≤ δ, i = 2,3, . . . ,k.
That is,
p si−δi ≤ ai−1p
q
2 ≤ p si+δi , i = 2,3, . . . ,k. (2)
It is clear that si →+∞ as q →+∞ (i = 2,3, . . . ,k). Now we prove that
((pi −1)p si−1i , p
si
i ) ⊆L . (3)
Let n ∈ ((pi −1)p si−1i , p
si
i ). Then p
si−1
i |Ln . Since
1+ 1
2
+·· ·+ 1
pi −1
= ∑
1≤ j≤(pi−1)/2
(
1
j
+ 1
pi − j
)
= ∑
1≤ j≤(pi−1)/2
pi
j (pi − j )
,
it follows that
Hn =
n∑
j=1,p si −1i - j
1
j
+ 1
p si−1i
(
1+ 1
2
+·· ·+ 1
pi −1
)
=
n∑
j=1,p si −1i - j
1
j
+ 1
p si−2i
∑
1≤ j≤(pi−1)/2
1
j (pi − j )
= a
b
+ 1
p si−2i
c
d
,
C. R. Mathématique — 2022, 360, 53-57
56 Bing-Ling Wu and Xiao-Hui Yan
where p si−1i - b and pi - d . Then νpi (Hn) ≥ −(si − 2). Hence, vn is not divisible by p
si−1
i , that is,
p si−1i - vn . Noting that vn |Ln and p
si−1
i |Ln , we have vn < Ln . Hence n ∈L . So (3) holds. It follows
from (3) that
L (pq2 ) ≥
k∑
i=2
|L ∩ (ai pq2 , ai−1p
q
2 )|
≥
k∑
i=2
|((pi −1)p si−1i , p
si
i )∩ (ai p
q
2 , ai−1p
q
2 )|
=
k∑
i=2
(
|(ai pq2 , ai−1p
q
2 )|− |(ai p
q
2 , ai−1p
q
2 ) \ ((pi −1)p si−1i , p
si
i )|
)
≥
k∑
i=2
(
ai−1p
q
2 −ai p
q
2 −1−|(ai p
q
2 , ai−1p
q
2 ) \ ((pi −1)p si−1i , p
si
i )|
)
= a1pq2 −ak p
q
2 −k +1−
k∑
i=2
|(ai pq2 , ai−1p
q
2 ) \ ((pi −1)p si−1i , p
si
i )|
≥ pq2 −
1
2
εpq2 −k −
k∑
i=2
|(ai pq2 , ai−1p
q
2 ) \ ((pi −1)p si−1i , p
si
i )|.
It follows from Lemma 5 that
|(ai pq2 , ai−1p
q
2 ) \ ((pi −1)p si−1i , p
si
i )| ≤ |ai p
q
2 − (pi −1)p si−1i |+ |ai−1p
q
2 −p sii |+2
=
(
1− 1
pi
)
|ai−1pq2 −p sii |+ |ai−1p
q
2 −p sii |+2
≤ 2|ai−1pq2 −p sii |+2.
If ai−1p
q
2 ≥ p sii , then by (2) we have
0 ≤ ai−1pq2 −p sii ≤ p
si+δ
i −p
si
i =
(
pδi −1
)
p sii .
If ai−1p
q
2 < p sii , then by (2) we have
0 ≤ p sii −ai−1p
q
2 ≤ p sii −p
si−δ
i = p−δi
(
pδi −1
)
p sii ≤
(
pδi −1
)
p sii .
In all cases, we have
|ai−1pq2 −p sii | ≤
(
pδi −1
)
p sii .
By (2),
p sii ≤ ai−1p
q
2 p
δ
i ≤ p
q
2 p
δ
i .
Hence,
|ai−1pq2 −p sii | ≤
(
pδi −1
)
pδi p
q
2 .
In view of (1),
0 < δ log pi ≤ δ log pk <
ε
16k
< 1
16
.
It follows from ex −1 ≤ 2x (0 ≤ x ≤ 12 ) that(
pδi −1
)
pδi ≤ 2δ(log pi )eδ log pi <
ε
8k
e1/16 < ε
4k
.
Thus,
|ai−1pq2 −p sii | <
ε
4k
pq2 .
It follows that
k∑
i=2
|(ai pq2 , ai−1p
q
2 ) \ ((pi −1)p si−1i , p
si
i )| < k
( ε
2k
pq2 +2
)
= 1
2
εpq2 +2k.
C. R. Mathématique — 2022, 360, 53-57
Bing-Ling Wu and Xiao-Hui Yan 57
Hence,
L (pq2 ) ≥ p
q
2 −
1
2
εpq2 −k −
1
2
εpq2 −2k = p
q
2 −εp
q
2 −3k.
Thus,
d̄(L ) ≥ lim
q→∞
L (pq2 )
pq2
≥ 1−ε,
where q →∞ and q satisfying (2). Therefore, d̄(L ) = 1. This completes the proof.
Acknowledgments
We would like to thank the referee for his/her helpful comments.
References
[1] D. W. Boyd, “A p-adic study of the partial sums of the harmonic series”, Exp. Math. 3 (1994), no. 4, p. 287-302.
[2] A. Eswarathasan, E. Levine, “p-integral harmonic sums”, Discrete Math. 91 (1991), no. 3, p. 249-257.
[3] G. H. Hardy, E. M. Wright, An introduction to the theory of numbers, 5th ed., Oxford University Press, 1979.
[4] S. Lang, Introduction to transcendental numbers, Addison-Wesley Series in Mathematics, Addison-Wesley Publishing
Group, 1966.
[5] C. Sanna, “On the p-adic valuation of harmonic numbers”, J. Number Theory 166 (2016), p. 41-46.
[6] P. Shiu, “The denominators of harmonic numbers”, https://arxiv.org/abs/1607.02863v1, 2016.
[7] B.-L. Wu, Y.-G. Chen, “On certain properties of harmonic numbers”, J. Number Theory 175 (2017), p. 66-86.
[8] ——— , “On the denominators of harmonic numbers”, C. R. Math. Acad. Sci. Paris 356 (2018), no. 2, p. 129-132.
[9] ——— , “On the denominators of harmonic numbers. II”, J. Number Theory 200 (2019), p. 397-406.
C. R. Mathématique — 2022, 360, 53-57


On the denominators of harmonic numbers. IV

Peng Chen

Ji Li

Lesley Ward

Lixin Yan

Weak-type endpoint bounds for Bochner–Riesz means for the Hermite operator

Nguyen Tien Dung

Nguyen Thu Hang

Pham Thi Phuong Thuy

Comptes Rendus
Mathématique
Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy
Density estimates for the exponential functionals of fractional Brownian
motion
Volume 360 (2022), p. 151-159
<https://doi.org/10.5802/crmath.274>
© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2022, 360, p. 151-159
https://doi.org/10.5802/crmath.274
Probability theory / Probabilités
Density estimates for the exponential
functionals of fractional Brownian motion
Nguyen Tien Dung∗, a, Nguyen Thu Hangb and Pham Thi Phuong Thuyc
a Department of Mathematics, VNU University of Science, Vietnam, National
University, Hanoi, 334 Nguyen Trai, Thanh Xuan, Hanoi, 084 Vietnam
b Department of Mathematics, Hanoi University of Mining and Geology, 18 Pho Vien,
Bac Tu Liem, Hanoi, 084 Vietnam
c The faculty of Basic Sciences, Vietnam Air Defence and Air Force Academy, Son Tay,
Ha Noi, 084 Vietnam
E-mails: dung@hus.edu.vn (N. Tien Dung), thuhangmdc@gmail.com (N. Thu Hang),
thuypham82.com@gmail.com (P. T. Phuong Thuy)
Abstract. In this note, we investigate the density of the exponential functional of the fractional Brownian
motion. Based on the techniques of Malliavin’s calculus, we provide a log-normal upper bound for the density.
Mathematical subject classification (2010). 60G22, 60H07.
Funding. This research was funded by the Vietnam National University, Hanoi under grant number QG.20.21.
Manuscript received 6th March 2021, revised 19th June 2021, accepted 21st September 2021.
1. Introduction
Let B H = (B Ht )t ∈ [0,T ] be a fractional Brownian motion (fBm) with Hurst index H ∈ (0,1). We recall
that B H is a centered Gaussian process with covariance function
RH (t , s) := E
[
B Ht B
H
s
]
= 1
2
(
t 2H + s2H −|t − s|2H
)
, 0 ≤ s, t ≤ T.
We consider the exponential functional of the form
F =
∫ T
0
eas+σB
H
s d s, (1)
where T > 0, a ∈ R and σ > 0 are constants. It is known that this functional plays an important
role in several domains. The special case, where H = 12 , has been well studied and a lot of fruitful
properties of F can be founded in the literature, see e.g. [3, 4, 10]. However, to the best our
knowledge, the deep properties of F for H 6= 12 are scarce. In a recent paper [8], we have proved the
Lipschitz continuity of the cumulative distribution function of F with respect to the Hurst index
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
152 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy
H . The aim of the present paper is to investigate the density of F. Unlike the case H = 12 , it is not
easy to find the density of F explicitly for H 6= 12 and hence, our work will focus on providing the
estimates for the density function. It should be noted that, in the last years, the density estimates
for random variables related to fBm has been extensively studied, see e.g. [1,2,6,7] and references
therein.
The rest of this article is organized as follows. In Section 2, we briefly recall some of the relevant
elements of the Malliavin calculus and two general estimates for densities. Our main results are
then stated and proved in Section 3. Our Theorems 6 and 8 point out that the density of F is
bounded from above by log-normal densities.
2. Preliminaries
In the whole paper, we assume H > 12 . Under this assumption, fBm admits the Volterra represen-
tation
B Ht =
∫ t
0
K (t , s)dBs , (2)
where (Bt )t∈[0,T ] is a standard Brownian motion and for some normalizing constant cH , the kernel
K is given by
K (t , s) = cH s1/2−H
∫ t
s
(u − s)H− 32 uH− 12 du, 0 < s ≤ t ≤ T.
Let us recall some elements of Malliavin calculus with respect to Brownian motion B , where B
is used to present B Ht as in (2). We suppose that (Bt )t ∈ [0,T ] is defined on a complete probability
space (Ω,F ,F,P ), where F= (Ft )t ∈ [0,T ] is a natural filtration generated by the Brownian motion
B. For h ∈ L2[0,T ], we denote by B(h) the Wiener integral
B(h) =
∫ T
0
h(t )dBt .
Let S denote the dense subset of L2(Ω,F ,P ) consisting of smooth random variables of the form
F = f (B(h1), . . . , B(hn)), (3)
where n ∈ N, f ∈ C∞b (Rn),h1, . . . , hn ∈ L2[0,T ]. If F has the form (3), we define its Malliavin
derivative as the process DF := {D t F, t ∈ [0,T ]} given by
D t F =
n∑
k=1
∂ f
∂xk
(
B(h1), . . . , B(hn)
)
hk (t ).
More generally, for each k ≥ 1, we can define the iterated derivative operator by setting
Dkt1, ..., tk F = D t1 . . . D tk F.
For any p,k ≥ 1, we shall denote by Dk,p the closure of S with respect to the norm
‖F‖pk, p := E |F |p +E
[∫ T
0
∣∣D t1 F ∣∣p d t1]+·· ·+E [∫ T
0
. . .
∫ T
0
∣∣∣Dkt1, ..., tk F ∣∣∣p d t1 . . . d tk] .
A random variable F is said to be Malliavin differentiable if it belongs to D1,2. For any F ∈ D1,2,
the Clark–Ocone formula says that
F −E [F ] =
∫ T
0
E
[
Ds F
∣∣Fs]dBs .
Moreover, any F,G ∈D1,2, we have the following covariance formula
Cov(F,G) = E
[∫ T
0
Ds F E
[
DsG
∣∣Fs]d s] . (4)
In order to obtain the density estimates for exponential functionals we need the following general
results.
C. R. Mathématique — 2022, 360, 151-159
Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 153
Proposition 1. Let q,α,β be three positive real numbers such that 1q + 1α+ 1β = 1. Let F be a random
variable in the spaceD2,α, such that E [‖DF‖−2βH ] <∞. Then the density ρF (x) of F can be estimated
as follows
ρF (x) ≤ cq,α,β (P (F ≤ x))1/q ×
(
E
[
‖DF‖−1H
]
+∥∥D2F∥∥Lα(Ω;H⊗H) ∥∥‖DF‖−2H ∥∥β) , x ∈R, (5)
where cq,α,β is a positive constant and H = L2[0,T ].
Proof. This proposition comes from the computations on [5, p. 87]. 
Proposition 2. Let F ∈D2,4 be such that E [F ] = 0. Define the random variable
ΦF :=
∫ T
0
Ds F E
[
Ds F
∣∣Fs]d s.
Assume that ΦF 6= 0 a.s. and the random variables FΦF and
1
Φ2F
∫ T
0 DsΦF E [Ds F |Fs ]d s belong to
L2(Ω). Then the law of F has a continuous density given by
ρF (x) = ρF (0)exp
(
−
∫ x
0
hF (z)d z
)
exp
(
−
∫ x
0
wF (z)d z
)
, x ∈ suppρF , (6)
where the functions wF and hF are defined by
wF (z) := E
[
F
ΦF
∣∣∣∣F = z] , hF (z) := E
[
1
Φ2F
∫ T
0
DsΦF E
[
Ds F
∣∣Fs]d s
∣∣∣∣∣F = z
]
.
Proof. This proposition is Theorem 7 in our recent paper [9]. 
3. The main results
In this section, we provide explicit estimates for the density ρF (x) of the functional F defined
by (1). Our idea is to consider the random variable X := lnF − E [lnF ] and use the relation
ρF (x) = 1x ρX (ln x −E [lnF ]), x > 0, where ρX denotes the density of X .
We need some technical results.
Proposition 3. Consider the random variable X := lnF −E [lnF ]. It holds that
0 ≤ DθX ≤σK (T,θ) a.s. (7)
0 ≤ Dr DθX ≤ 2σ2K (T,θ)K (T,r ) a.s. (8)
Proof. By the chain rule for Malliavin derivatives, we have, for 0 ≤ r,θ ≤ T,
DθX =
σ
∫ T
θ K (s,θ)e
as+σB Hs d s∫ T
0 e
as+σB Hs d s
(9)
and
Dr DθX =
σ2
∫ T
θ∨r K (s,θ)K (s,r )e
as+σB Hs d s∫ T
0 e
as+σB Hs d s
− σ
2
∫ T
r K (s,r )e
as+σB Hs d s
∫ T
θ K (s,θ)e
as+σB Hs d s(∫ T
0 e
as+σB Hs d s
)2 .
Because the function s 7→ K (s,θ) is non-decreasing for each θ, (7) follows directly from (9). We
also have
Dr DθX ≤ 2σ2K (T,θ)K (T,r ) a.s.
To prove the non-negativity of the second order Malliavin derivative, we let U be a random
variable with the density function defined by
f (x) = e
ax+σB Hx∫ T
0 e
as+σB Hs d s
, 0 ≤ x ≤ T.
C. R. Mathématique — 2022, 360, 151-159
154 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy
Denote by EU the expectation with respect to U . We have
Dr DθX = EU [K (U ,θ)K (U ,r )]−EU [K (U ,θ)]EU [K (U ,r )] .
Note that the functions s 7→ K (s,θ) and s 7→ K (s,r ) are non-decreasing. Hence, by Chebyshev’s
association inequality, Dr DθX ≥ 0 a.s. The proof of Proposition 3 is complete. 
Lemma 4. Define
Mr := E [F |Fr ] = E
[∫ T
0
eas+σk f l6B
H
s d s
∣∣∣∣Fr ] , 0 ≤ r ≤ T.
Then, for every p ≥ 2, we have
E
[(
max
0≤r ≤T
Mr
)p]
≤C <∞,
where C is a positive constant depending on p,T, a,σ and H .
Proof. The stochastic process M := (Mr )0≤r≤T is a martingale with M0 = E [F ] and MT = F. Hence,
by Burkhölder–David–Gundy inequality, we have
E
[(
max
0≤r ≤T
Mr
)p]
≤ cp
(
M p0 +E
[
〈M〉p/2T
])
= cp
(
(E [F ])p +E
[
〈M〉p/2T
])
, (10)
where cp is a positive constant. Using the Clark–Ocone formula we have
MT = E MT +
∫ T
0
E
[
Dr MT
∣∣Fr ]dBr
= E [F ]+σ
∫ T
0
E
[∫ T
r
K (s,r )eas+σB
H
s d s
∣∣∣∣Fr ]dBr ,
which gives us
〈M〉T =
∫ T
0
σ2
(
E
[∫ T
r
K (s,r )eas+σB
H
s d s
∣∣∣∣Fr ])2 dr
≤
∫ T
0
σ2K 2(T,r )M 2r dr a.s.
Then, by Hölder inequality, we have
E
[
〈M〉p/2T
]
≤σp E
[(∫ T
0
K
2p−4
p (T,r )K
4
p (T,r )M 2r dr
)p/2]
≤σp
(∫ T
0
K 2(T,r )dr
) p
2 −1 (∫ T
0
K 2(T,r )E
[
M pr
]
dr
)
≤σp T (p−2)H
(∫ T
0
K 2(T,r )E
[
F p
]
dr
)
=σp T pH E[F p].
(11)
Here we used the fact that
∫ T
0 K
2(T,r )dr = E |B HT |2 = T 2H . So we obtain the desired conclusion by
inserting (11) into (10). 
Proposition 5. Let X be as in Proposition 3. We defineΦX :=
∫ T
0 Ds X E [Ds X |Fs ]d s. Then,∣∣ΦX ∣∣−1 ∈ Lp (Ω), ∀ p ≥ 1.
We also have (∫ T
0
∣∣DθX ∣∣2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1.
C. R. Mathématique — 2022, 360, 151-159
Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 155
Proof. It follows from (9) that
DθX ≥
σ
T
e
−2|a|T+σ min
0≤ s≤T
B Hs −σ max
0≤ s≤T
B Hs
∫ T
θ
K (s,θ)d s a.s. (12)
On the other hand, by using the Cauchy–Schwarz inequality, we have
E
[
DθX
∣∣Fθ]≥ σ
(
E
[√∫ T
θ K (s,θ)e
as+σB Hs d s
∣∣∣∣Fθ])2
E
[∫ T
0 e
as+σB Hs d s
∣∣∣Fθ] a.s.
and √∫ T
θ
K (s,θ)eas+σB Hs d s ≥
∫ T
θ K (s,θ)
√
eas+σB Hs d s√∫ T
θ K (s,θ)d s
a.s.
We therefore get
E
[
DθX
∣∣Fθ]≥ σ
(∫ T
θ K (s,θ)E
[
eas/2+σB
H
s /2
∣∣∣Fθ]d s)2∫ T
θ K (s,θ)d sE
[∫ T
0 e
as+σB Hs d s
∣∣∣Fθ] a.s.
Furthermore, by Lyapunov’s inequality,
E
[
eas/2+σB
H
s /2
∣∣∣Fθ]≥ eas/2+σE[B Hs ∣∣Fθ]/2 a.s.
As a consequence,
E
[
DθX
∣∣Fθ]≥ σe−|a|T+σ min0≤θ≤ s≤T Ns,θ ∫ Tθ K (s,θ)d smax
0≤θ≤T
Mθ
a.s. (13)
where
Ns,θ := E
[
B Hs
∣∣∣Fθ] and Mθ := E [∫ T
0
eas+σB
H
s d s
∣∣∣∣Fθ] .
Combining (12) and (13) yields
DθX E
[
DθX
∣∣Fθ]≥ σ2T e−3|a|T+σ min0≤ s≤T B Hs −σ max0≤ s≤T B Hs +σ min0≤θ≤ s≤T Ns,θ
(∫ T
θ K (s,θ)d s
)2
max
0≤θ≤T
Mθ
a.s.
and hence,
ΦX ≥ σ
2
T
e
−3|a|T+σ min
0≤ s≤T
B Hs −σ max
0≤ s≤T
B Hs +σ min
0≤θ≤ s≤T
Ns,θ
∫ T
0
(∫ T
θ K (s,θ)d s
)2
dθ
max
0≤θ≤T
Mθ
a.s. (14)
We observe that ∫ T
0
(∫ T
θ
K (s,θ)d s
)2
dθ =
∫ T
0
∫ T
θ
∫ T
θ
K (t ,θ)K (s,θ)d sd tdθ
=
∫ T
0
∫ T
0
(∫ s∧t
0
K (t ,θ)K (s,θ)dθ
)
d sd t
=
∫ T
0
∫ T
0
E
[
B Ht B
H
s
]
d sd t = T
2H+2
2H +2 .
This, together with (14), yields∣∣ΦX ∣∣−1 ≤ 2H +2
σ2T 2H+1
e
3|a|T+2σ max
0≤ s≤T
B Hs +σ max
0≤θ≤ s≤T
Ns,θ
max
0≤θ≤T
Mθ a.s.
C. R. Mathématique — 2022, 360, 151-159
156 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy
We have (Ns,θ)0≤θ≤ s≤T is a Gaussian field with finite variances because Ns,θ =
∫ θ
0 KH (s,r )dBr .
Hence, by Fernique’s theorem, there exists ε> 0 such that
E
[
e
ε max
0≤θ≤ s≤T |Ns,θ|
2]
<∞.
Since
e
pσ max
0≤θ≤ s≤T
Ns,θ ≤ e
p2σ2
4ε +ε max0≤θ≤ s≤T |Ns,θ|
2
,
this implies that
e
σ max
0≤θ≤ s≤T
Ns,θ ∈ Lp (Ω) for any p ≥ 1.
Similarly, we also have
e
2σ max
0≤ s≤T
B Hs ∈ Lp (Ω) for any p ≥ 1.
So, recalling Lemma 4, we conclude that |ΦX |−1 ∈ Lp (Ω) for any p ≥ 1.
We deduce from (12) that∫ T
0
∣∣DθX ∣∣2dθ ≥ σ2
(2H +2)T 2H e
−4|a|T+2σ min
0≤ s≤T
B Hs −2σ max
0≤ s≤T
B Hs a.s. (15)
Hence, we also have (
∫ T
0 |DθX |2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1. The proof of Proposition 5 is complete.

We now are in a position to bound the density ρF (x) of F. We first use Proposition 1 to estimate
the left tail of the density.
Theorem 6. We have
ρF (x) ≤ c
x
exp
(
− (ln x −E [lnF ])
2
8σ2T 2H
)
, 0 < x ≤ e E [lnF ], (16)
where c is a positive constant.
Proof. It is known from Proposition 5 that
‖D X ‖−2H =
(∫ T
0
∣∣DθX ∣∣2dθ)−1 ∈ Lp (Ω), ∀ p ≥ 1.
In addition, from the estimate (8), we have
∥∥D2X ∥∥2L2(Ω ; H⊗H) = ∫ T
0
∫ T
0
E
∣∣DθDr X ∣∣2dθdr ≤σ4 ∫ T
0
∫ T
0
K 2(T,θ)K 2(T,r )dθdr =σ4T 2H <∞.
The above estimates allow us to use Proposition 1 with q =β= 4,α= 2 and we obtain
ρX (x) ≤ cP (X ≤ x)
1
4 , x ∈R, (17)
where c is a positive constant.
The remaining of the proof is to bound P (X ≤ x) for x ≤ 0. We consider the function ϕ(λ
:= E [e−λX ], λ > 0 (this function is well defined because F−1 ∈ Lp (Ω), ,∀ p ≥ 1). By using
repeatedly the covariance formula (4), we have
C. R. Mathématique — 2022, 360, 151-159
Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 157
σ2X :=Var(X ) = E
[
ΦX
]
and
ϕ′(λ) =−E
[
X e−λX
]
=λE
[
e−λXΦX
]
=λσ2X E
[
e−λX
]
+λE
[
e−λX
(
ΦX −σ2X
)]
=λσ2X E
[
e−λX
]
−λ2E
[
e−λX
∫ T
0
Ds X E
[
DsΦX
∣∣Fs]d s]
Since Ds X ≥ 0 and Dr Ds X ≥ 0, those imply that
∫ T
0 Ds X E [DsΦX |Fs ]d s ≥ 0, and hence,
ϕ′(λ) ≤λσ2X E
[
e−λX
]
=λσ2Xϕ(λ), λ> 0.
This, together the fact ϕ(0) = 1, gives us
ϕ(λ) ≤ e
λ2σ2X
2 , λ> 0.
By Markov’s inequality we have, for all λ> 0,
P (X ≤ x) ≤ eλxϕ(λ) ≤ eλx+
λ2σ2X
2 , x ≤ 0.
When x ≤ 0, we can choose λ=− x
σ2X
to get
P (X ≤ x) ≤ e
− x2
2σ2X , x ≤ 0.
From the estimate (7), we have σ2X = E [ΦX ] ≤
∫ T
0 |Ds X |2d s ≤σ2T 2H . So we deduce
P (X ≤ x) ≤ e−
x2
2σ2T 2H , x ≤ 0. (18)
Combining (17) and (18) yields
ρX (x) ≤ c e−
x2
8σ2T 2H , x ≤ 0.
where c is a positive constant. Recalling X = lnF − E [lnF ], the density of F satisfies ρF (x) =
1
x ρX (ln x −E [lnF ]). When 0 < x ≤ eE [lnF ], we have y := ln x −E [lnF ] ≤ 0. We thus obtain
ρF (x) = 1
x
ρX (y) ≤ c
x
e
− y2
8σ2T 2H = c
x
e
− (ln x−E [lnF ])2
8σ2T 2H , 0 < x ≤ e E [lnF ].
This completes the proof of Theorem 6. 
Remark 7. Replacing X by F −E [F ] in the proof of Theorem 6, we obtain the following Gaussian
bound for the left tail
ρF (x) ≤ c e
− (x−E [F ])2
8σ2F , x ≤ E [F ],
where σ2F := Var(F ) and c is a positive constant.
We now use Proposition 2 to estimate the right tail of the density.
Theorem 8. We have
ρF (x) ≤ c
x
exp
(
− (ln x −E [lnF ])
2
2σ2T 2H
)
, x > eE [lnF ], (19)
where c is a positive constant.
C. R. Mathématique — 2022, 360, 151-159
158 Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy
Proof. Let X be as in Proposition 3. Obviously, we have ΦX 6= 0 a.s. Moreover, from the esti-
mates (7) and (8) we obtain
0 ≤ DsΦX =
∫ T
0
Ds DθX E
[
DθX
∣∣Fθ]dθ+∫ T
0
DθX E
[
Ds DθX
∣∣Fθ]dθ
≤ 4σ3
∫ T
0
K 2(T,θ)K (T, s)dθ = 4σ3K (T, s)T 2H
and
0 ≤
∫ T
0
DsΦX E
[
Ds X
∣∣Fs]d s ≤ 4σ4T 2H ∫ T
0
K 2(T, s)d s = 4σ4T 4H
Hence, it follows from Proposition 5 that the random variable 1
Φ2X
∫ T
0 DsΦX E [Ds X |Fs ]d s belong
to L2(Ω). We also have XΦX ∈ L2(Ω) because
−|a|T −σ max
0≤ s≤T
B Hs + lnT −E [lnF ] ≤ X ≤ |a|T +σ max
0≤ s≤T
B Hs + lnT −E [lnF ]
and hence, X ∈ Lp (Ω) for all p ≥ 2.
In view of Proposition 2, the density ρX (x) of X is given by
ρX (x) = ρX (0)exp
(
−
∫ x
0
hX (z)d z
)
exp
(
−
∫ x
0
wX (z)d z
)
, x ∈ supp ρX , (20)
where
wX (z) := E
[
X
ΦX
∣∣∣∣X = z] and hX (z) := E
[
1
Φ2X
∫ T
0
DsΦX E
[
Ds X
∣∣Fs]d s
∣∣∣∣∣X = z
]
.
Since hX ≥ 0, this implies that
exp
(
−
∫ x
0
hX (z)d z
)
≤ 1, x ≥ 0.
From the estimate (7) we have
0 ≤ΦX ≤σ2
∫ T
0
K 2(T,θ)dθ =σ2T 2H a.s.
and we obtain
exp
(
−
∫ x
0
wF (z)d z
)
≤ e−
x2
2σ2T 2H , x ∈R.
So we can conclude that
ρX (x) ≤ ρX (0)e−
x2
2σ2T 2H , x ≥ 0,
and (19) follows because ρF (x) = 1x ρX (ln x −E [lnF ]). The proof of Theorem 8 is complete. 
Acknowledgments
The author would like to thank the anonymous referee for their valuable comments for improving
the paper. A part of this paper was done while the first author was visiting the Vietnam Institute
for Advanced Study in Mathematics (VIASM). He would like to thank the VIASM for financial
support and hospitality.
C. R. Mathématique — 2022, 360, 151-159
Nguyen Tien Dung, Nguyen Thu Hang and Pham Thi Phuong Thuy 159
References
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fractional Brownian motions”, Ann. Probab. 44 (2016), no. 1, p. 399-443.
[2] J. Liu, C. A. Tudor, “Stochastic heat equation with fractional Laplacian and fractional noise: existence of the solution
and analysis of its density”, Acta Math. Sci. 37 (2017), no. 6, p. 1545-1566.
[3] H. Matsumoto, M. Yor, “Exponential functionals of Brownian motion. I. Probability laws at fixed time”, Probab. Surv.
2 (2005), p. 312-347.
[4] ——— , “Exponential functionals of Brownian motion. II. Some related diffusion processes”, Probab. Surv. 2 (2005),
p. 348-384.
[5] D. Nualart, The Malliavin calculus and related topics, second ed., Probability and Its Applications, Springer, 2006.
[6] N. V. Tan, “Smoothness and Gaussian density estimates for stochastic functional differential equations with frac-
tional noise”, Stat. Optim. Inf. Comput. 8 (2020), no. 4, p. 822-833.
[7] N. Tien Dung, “The density of solutions to multifractional stochastic Volterra integro-differential equations”, Non-
linear Anal., Theory Methods Appl. 130 (2016), p. 176-189.
[8] ——— , “Kolmogorov distance between the exponential functionals of fractional Brownian motion”, C. R. Math.
Acad. Sci. Paris 357 (2019), no. 7, p. 629-635.
[9] ——— , “Gaussian lower bounds for the density via Malliavin calculus”, C. R. Math. Acad. Sci. Paris 358 (2020), no. 1,
p. 79-88.
[10] M. Yor, Exponential functionals of Brownian motion and related processes, Springer Finance, Springer, 2001.
C. R. Mathématique — 2022, 360, 151-159


Density estimates for the exponential functionals of fractional Brownian motion

Charles-Edouard Bréhier

The averaging principle for stochastic differential equations driven by a Wiener process revisited

Yong Yang

Comptes Rendus
Mathématique
Yong Yang
On the number of prime divisors of character degrees and conjugacy
classes of a finite group
Volume 360 (2022), p. 583-588
<https://doi.org/10.5802/crmath.301>
© Académie des sciences, Paris and the authors, 2022.
Some rights reserved.
This article is licensed under the
Creative Commons Attribution 4.0 International License.
http://creativecommons.org/licenses/by/4.0/
Les Comptes Rendus. Mathématique sont membres du
Centre Mersenne pour l’édition scientifique ouverte
www.centre-mersenne.org
Comptes Rendus
Mathématique
2022, 360, p. 583-588
https://doi.org/10.5802/crmath.301
Algebra / Algèbre
On the number of prime divisors of character
degrees and conjugacy classes of a finite group
Yong Yang∗, a, b
a Three Gorges Mathematical Research Center, College of Science, China Three
Gorges University, Yichang, Hubei 443002, China
b Department of Mathematics, Texas State University, San Marcos, TX 78666, USA
E-mail: yang@txstate.edu
Abstract. A result of Gluck is that any finite group G has an abelian subgroup A such that |G : A| is bounded
by a polynomial function of the largest irreducible character degree of G . Moretó presented a variation of this
result that looks at the number of prime factors of the irreducible character degrees and obtained an almost
quadratic bound. The author improved the result of Moretó to almost linear. In this note, we further improve
the bound, and also study the related problem on conjugacy class sizes.
Mathematical subject classification (2010). 20C15, 20C20.
Funding. This work was partially supported by the Simons Foundation (No 499532).
Manuscript received 28th September 2021, accepted 25th November 2021.
1. Introduction
Let G be a finite group, F(G) be the Fitting subgroup of G , and denote by b(G) = max{ψ(1) | ψ ∈
Irr(G)} the largest degree of an irreducible character of G . Gluck proved in [6] that any finite group
G has an abelian subgroup A such that |G : A| is bounded by a polynomial function of b(G). For
solvable groups, Gluck showed that |G : F(G)| ≤ b(G)13/2 and conjectured that |G : F(G)| ≤ b(G)2.
This is a well-known conjecture for solvable groups.
In more recent work, Moretó [8] explored a different notion of “large”. The obvious meaning
for the word “large” refers to the absolute value of an integer n. The meaning that Moretó
considered involves the prime factorization of n; n is “large” if it has many prime divisors
(counting multiplicities). Given an integer n = pa11 · · ·patt as a product of powers of different
primes, we define ω(n) = a1 + ·· · + an and ωpi (n) = ai . Let G be a finite group, we use Irr(G)
to denote the set of complex irreducible characters of G and cl(G) to denote the set of conjugacy
classes of G . We set ω(G) = max{ω(χ(1)) |χ ∈ Irr(G)} and ωcl (G) = max{ω(|C |) |C ∈ cl(G)}.
Moretó [8] proved the following variation of the theorem of Gluck. In this paper, the base of
the log will always be 2.
∗Corresponding author.
ISSN (electronic) : 1778-3569 https://comptes-rendus.academie-sciences.fr/mathematique/
584 Yong Yang
Theorem 1. There exist (universal) constants D1 and D2 such that if G is a non-abelian finite
group then G has an abelian subgroup A satisfying
ω(|G : A|) ≤ D1ω(G)2 logω(G)+D2.
Moretó speculated that a linear bound might exist. In [12], the author improved the bound in
Theorem 1 to almost linear.
Theorem 2. There exists a (universal) constant D such that G has an abelian subgroup A satisfying
ω(|G : A|) ≤ Dω(G) logω(G) when ω(G) > 1.
In this paper, we first strengthen Theorem 2.
Theorem 3. There exist (universal) constants D3 and D4 such that G has an abelian subgroup A
satisfying ω(|G : A|) ≤ D3ω(G)(loglog(ω(G))+D4) when ω(G) > 1.
After this, we also study the related question about conjugacy classes.
Theorem 4. There exist (universal) constants D5 and D6 such that ω(|G : F(G)|) ≤ D5ωcl (G)
(loglog(ωcl (G))+D6) when ωcl (G) > 1.
2. Preliminary Results
Lemma 5. Let N be a normal subgroup of G.
(1) If x ∈ N , then |xN | divides |xG |.
(2) If x ∈G, then |(xN )G/N | divides |xG |.
Lemma 6. Let G be a finite simple group. Then ω(|Aut(G)|) ≤ 2ω(|G|).
Proof. This is [8, Lemma 2.7]. 
Lemma 7. Let n ≥ 2 be a positive integer. Thenω(n!) ≤ 2n(log(logn)+C ) for some fixed constant C .
Proof. This is a standard result in number theory. We have
ωp (n!) =
⌊
logp n
⌋∑
k=1
⌊
n
pk
⌋
≤
∞∑
k=1
⌊
n
pk
⌋
≤ n
p −1 ≤
2n
p
.
Hence by [5], we have
ω(n!) ≤ 2n ∑
p ≤n
1
p
≤ 2n · (loglogn +C) .

Lemma 8. There exist (universal) constants C1 and C2 such that if G is a nonabelian simple group,
then ω(|G|) ≤C1ω(G)(loglogω(G)+C2).
Proof. Let G be a finite simple group which is not an alternating group, then ω(|G|) ≤ 5ω(G)
by [8, Lemmas 2.2, 2.3, 2.4].
Let G be an alternating group of degree n ≥ 5. Note that ω(|G|) =ω(n!/2). Since ω2(n!) ≥ n2 , we
have 2ω(G) ≥ω2(n!) ≥ n2 by [4, Lemma 1.2] (for the exceptional cases in the table one may verify
this by a direct calculation). By Lemma 7, we have
ω(|G|) ≤ 8ω(G) · (log(log4ω(G))+C) .
We note that ω(G) ≥ 2 since G is a nonabelian simple group. Thus,
ω(|G|) ≤ 8ω(G) · (log(logω(G))+C +2) .
We may choose C1 = 8, C2 =C +2 and the result follows. 
C. R. Mathématique — 2022, 360, 583-588
Yong Yang 585
Lemma 9. The alternating groups Altn of degree n ≥ 5 has conjugacy classes cl1 and cl2 such that
|Altn | | |cl1||cl2|.
Proof. For n = 5,6, this can be checked by a direct calculation.
For the case when n ≥ 7, set α= (1,2, . . . ,n) and β= (1,2, . . . , n −2) if n is odd. Then,
1
2
∣∣αSymn ∣∣= 1
2
(n)!/(n)
∣∣∣ ∣∣∣αAltn ∣∣∣
and
1
2
∣∣βSymn ∣∣= 1
2
(n)!/(2(n −2))
∣∣∣ ∣∣∣βAltn ∣∣∣ .
Since 2(n −2)n | 12 (n)!, we have 12 (n)!||αAltn | · |βAltn |, and the result follows.
Set α= (1,2, . . . , n −1) and β= (1,2, . . . , n −3) if n is even. Then,
1
2
∣∣αSymn ∣∣= 1
2
(n)!/(n −1)
∣∣∣ ∣∣∣αAltn ∣∣∣
and
1
2
∣∣βSymn ∣∣= 1
2
(n)!/(3(n −3))
∣∣∣ ∣∣∣βAltn ∣∣∣ .
Since 3(n −3)(n −1) | 12 (n)!, we have 12 (n)!||αAltn | · |βAltn |, and the result follows. 
Theorem 10. Let S 6= 2F4(2)′ be a simple group of Lie type. There exist semisimple elements
s1, s2 ∈ S such that |sS1 ||sS2 | is divisible by |S|.
Proof. The case S = PSL2(5) can be checked directly using [2] for instance. For S = PSL2(q) with
q > 5, it is well known that S has a class of size q(q +1) and another of size q(q −1) (see [3]), and
we are done. We therefore may assume that S 6= PSL2(q) from now on.
Let G be a simple algebraic group of simply connected type and F : G → G be a Steinberg
endomorphism such that S = Ḡ =G/Z with G := GF and Z := Z(G). For g ∈G , let g denote by the
image of g in S under the natural projection τ : G → S.
Let g ∈ G and denote Cg the preimage of CG (g ) under τ. Following [1, § 4.2], we consider
the map Tg : Cg → Z by h 7→ [h, g ] := h−1g−1hg . It is easy to show that Tg is in fact a group
homomorphism, and hence it follows that∣∣CG (g )∣∣ |Z | = ∣∣Cg ∣∣= ∣∣CG (g )∣∣ ∣∣Im(Tg )∣∣ .
In particular, |CG (g )| divides |CG (g )|, and therefore, it suffices to prove that there exist semisimple
elements g1, g2 ∈G such that |S| is divisible by |CG (g1)||CG (g2)|.
First, we consider the case when G is of exceptional type. Centralizers of semisimple
elements in such group are well known, see [7] for instance. One then can easily find the
required semisimple elements g1 and g2 such that |CG (g1)||CG (g2)| divides |S|. As an example,
the group G = E8(q) has two elements whose centralizers have orders Φ1(q)Φ2(q)Φ3(q)Φ5(q)
and Φ1(q)Φ2(q)Φ6(q)Φ10(q), where Φk (q) denotes the kth cyclotomic polynomial evaluated at
q , and thus they satisfy the desired condition. It remains to consider classical groups of rank at
least 2. We will produce the required semisimple elements g1, g2 ∈G using known knowledge on
the structure of semisimple elements in finite classical groups.
Let S = PSLn(q) with n ≥ 3. Using [10, Lemma 3.2], we can find g1 ∈ G = SLn(q) such that
CGLn (q)(g1)
∼= GL1(qn) and |CG (g1)| = (qn −1)/(q−1). (Here the centralizer is in fact a Coxter torus
of G .) Also, we can find g2 so that |CG (g2)| = qn−1 −1.
Let S = PSUn(q) with n ≥ 3. By [10, Lemma 3.3], we know that the centralizer of a semisimple
element in GUn(q) is a direct product of groups of the forms GLm(q2k ) and GUm(q2k−1) for
k ∈ Z≥0. Therefore we may choose g1 and g2 in SUn(q) such that the orders of their centralizers
C. R. Mathématique — 2022, 360, 583-588
586 Yong Yang
are (qn +1)/(q +1) and (qn−2 +1)(q −1) when n is odd and qn−1 +1 and (qn −1)/(q +1) when n
is even.
Let S = PSp2n(q) or Ω2n+1(q) with n ≥ 2. By [9, Lemmas 2.2 and 2.4], there exist semisimple
elements g1, g2 ∈ G such that CG (g1) = GL1(qn) and CG (g2) = GU1(qn), so that |S| is divisible by
|CG (g1)||CG (g2)|.
Lastly, let S = PΩε2n(q) with n ≥ 4 and ε = ±. Then G = P(COε2n(q))0. By [9, Lemmas 2.3 and
2.5], there are g1, g2 ∈ G such that |CG (g1)| = qn − ε1 and |CG (g2)| = (qn−1 + ε1)(q +1). The proof
of Theorem 10 is complete. 
Theorem 11. Let S be a nonabelian simple group. There exist two elements s1, s2 ∈ S such that
|sS1 ||sS2 | is divisible by |S|.
Proof. The sporadic groups can be checked by using [2]. The other cases follow from Lemma 9
and Theorem 10. 
3. Main Results
We first improve the main result of [12].
Proposition 12. Assume that N/G where N = L1× . . . ×Lm is the product of m finite non-abelian
simple groups Li permuted transitively by G. Let K =⋂i NG (Li ). Then ω(|G|) ≤ 2m(loglogm+C )+
2ω(|N |).
Proof. We observe that G/K is a permutation group on m letters and K /N ≤ Out(L1) × . . . ×
Out(Lm). Then the result follows from Lemma 7 and Lemma 6. 
Proposition 13. Let G be a group with no non-trivial solvable normal subgroup. Then ω(|G|) ≤
(2+2C1)ω(F∗(G))(loglogω(F∗(G))+C2).
Proof. The generalized Fitting subgroup F∗(G) = E1 ×E2 × ·· ·×En is a direct product of n non-
abelian simple groups on which G acts faithfully by conjugation.
Let K0 = G , {E11,E12, . . . , E1m1 } be an orbit of K0 on the set of simple direct factors of F∗(G),
and set L1 = E11 ×E12 × ·· ·×E1m1 . Let K1 = CK0 (L1), {E21,E22, . . . , E2m2 } be an orbit of K1 on the
set of simple direct factors of F∗(G), and set L2 = E21 ×E22 × ·· · ×E2m2 . Let K2 = CK1 (L2), and
inductively, we may define L3,K3 . . . Lt ,Kt where Kt = 1.
Clearly F∗(G) = L1 × L2 × ·· · × Lt . We know that Ki−1/Ki acts transitively and faithfully on
Li where i = 1, . . . , t and Li ∩ Ki = 1. Thus by Proposition 12, we have that ω(|Ki−1/Ki |) ≤
2mi (loglogmi +C )+2ω(|Li |).
Thus
ω(|G|) ≤ω (|K0/K1|)+·· ·+ω (|Kt−1/Kt |) ≤ 2
t∑
i=1
mi
(
loglogmi +C
)+2 t∑
i=1
ω (|Li |) .
Since n = m1 +·· ·+mt , we have
ω(|G|) ≤ 2n (loglogn +C)+2ω(∣∣F∗(G)∣∣) .
Since F∗(G) is a direct product of n simple groups, ω(F∗(G)) ≥ n. By Lemma 8, we have
ω(|G|) ≤ (2+2C1)ω
(
F∗(G)
)(
loglogω
(
F∗(G)
)+C2) . 
We now prove Theorem 3.
Proof. Let T be the maximal solvable normal subgroup of G . Then there exists an abelian
subgroup A of T such that ω(|T : A|) ≤ 19ω(T ) ≤ 19ω(G) by [12, Theorem 1.3].
C. R. Mathématique — 2022, 360, 583-588
Yong Yang 587
Let Ḡ =G/T , and we have that
ω
(∣∣Ḡ∣∣)≤ (2+2C1)ω(F∗ (Ḡ))(loglogω(F∗ (Ḡ))+C2)≤ (2+2C1)ω(G)(loglogω(G)+C2)
by Proposition 13.
Thus ω(|G : A|) =ω(|Ḡ|)+ω(|T : A|) ≤ D3ω(G)(loglog(ω(G))+D4) for some fixed constants D3
and D4. 
We now study the related problem on class sizes. We would expect a similar result would hold.
However, as the extra-special group shows, one cannot have a bound by just moduloing out an
abelian subgroup. On the other hand, if we replace an abelian subgroup by the Fitting subgroup,
we will get something along the same line.
Proposition 14. Let G be a finite solvable group. Then ω(|G : F(G)|) ≤ 15ωcl (G).
Proof. Since G is solvable, then there exist 15 conjugacy classes cl1, . . . , cl15 such that |G : F(G)|
divides |cl1| · · · |cl15| by [11, Theorem 3.5]. Thus we have ω(|G : F(G)|) ≤ 15ωcl (G). 
Proposition 15. Let G be a group with no non-trivial solvable normal subgroup. Then ω(|G|) ≤
2ωcl (F
∗(G))(loglogωcl (F∗(G))+C2 +2).
Proof. The generalized Fitting subgroup F∗(G) = E1 ×E2 × ·· ·×En is a direct product of n non-
abelian simple groups on which G acts faithfully by conjugation.
Let K0 = G , {E11,E12, . . . , E1m1 } be an orbit of K0 on the set of simple direct factors of F∗(G),
and set L1 = E11 ×E12 × ·· · ×E1m1 . Let K1 = CK0 (L1), {E21,E22, . . . , E2m2 } be an orbit of K1 on the
set of simple direct factors of F∗(G), and set L2 = E21 ×E22 × ·· · ×E2m2 . Let K2 = CK1 (L2), and
inductively, we may define L3,K3 . . . Lt ,Kt where Kt = 1.
Clearly F∗(G) = L1 × L2 · · · × Lt . We know that Ki−1/Ki acts transitively and faithfully on Li
where i = 1, . . . , t and Li ∩Ki = 1. Thus by Proposition 12, we have that
ω (|Ki−1/Ki |) ≤ 2mi
(
loglogmi +C
)+2ω (|Li |) .
Thus
ω(|G|) ≤ω (|K0/K1|)+·· ·+ω (|Kt−1/Kt |) ≤ 2
t∑
i=1
mi
(
loglogmi +C
)+2 t∑
i=1
ω (|Li |) .
Since n = m1 +·· ·+mt , we have
ω(|G|) ≤ 2n (loglogn +C2)+2ω(∣∣F∗(G)∣∣) .
Since F∗(G) is a direct product of n simple groups, ωcl (F∗(G)) ≥ n. By Theorem 11, we have
ω(|F∗(G)|) ≤ 2ωcl (F∗(G)). Thus,
ω(|G|) ≤ 2ωcl
(
F∗(G)
)(
loglogωcl
(
F∗(G)
)+C2 +2) .

We now prove Theorem 4.
Proof. Let T be the maximal solvable normal subgroup of G . Then ω(|T : F(G)|) ≤ 15ωcl (T ) ≤
15ωcl (G) by Proposition 14.
Let Ḡ =G/T , and we have that
ω(|Ḡ|) ≤ 2ωcl
(
F∗
(
Ḡ
))(
loglogωcl
(
F∗
(
Ḡ
))+C2 +2)≤ 2ωcl (G)(loglogωcl (G)+C2 +2)
by Proposition 15.
Thus ω(|G : F(G)|) = ω(|Ḡ|) +ω(|T : F(G)|) ≤ D5ωcl (G)(loglog(ωcl (G)) + D6) for some fixed
constants D5 and D6. 
C. R. Mathématique — 2022, 360, 583-588
588 Yong Yang
Acknowledgement
The author is grateful to the referee for the valuable suggestions which greatly improved the
manuscript.
References
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∑
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[6] D. Gluck, “The largest irreducible character degree of a finite group”, Can. J. Math. 37 (1985), p. 442-451.
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[8] A. Moretó, “A variation on theorems of Jordan and Gluck”, J. Algebra 301 (2006), no. 1, p. 274-279.
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C. R. Mathématique — 2022, 360, 583-588


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