On the Lehmer conjecture and counting in finite fields
We give a reformulation of the Lehmer conjecture about algebraic integers in terms of a simple counting problem modulo p.
math.NT↗
arXiv subjects
Emmanuel Breuillard
Publications and source records attributed to Emmanuel Breuillard.
On the dimension of Bernoulli convolutions
The Bernoulli convolution with parameter $λ\in(0,1)$ is the probability measure $μ_λ$ that is the law of the random variable $\sum_{n\ge0}\pmλ^n$, where the signs are independent unbiased coin tosses. We prove that each parameter $λ\in(1/2,1)$ with $\dimμ_λ<1$ can be approximated by algebraic parameters $ξ\in(1/2,1)$ within an error of order $\exp(-deg(ξ)^{A})$ for any number $A$, such that $\dimμ_ξ<1$. As a corollary, we conclude that $\dimμ_λ=1$ for each of $λ=\ln 2, e^{-1/2}, π/4$. These are the first explicit examples of such transcendental parameters. Moreover, we show that Lehmer's conjecture implies the existence of a constant $a<1$ such that $\dimμ_λ=1$ for all $λ\in(a,1)$.
math.CA↗
Expansion in simple groups
Two short seminal papers of Margulis used Kazhdan's property $(T)$ to give, on the one hand, explicit constructions of expander graphs, and to prove, on the other hand, the uniqueness of some invariant means on compact simple Lie groups. These papers opened a rich line of research on expansion and spectral gap phenomena in finite and compact simple groups. In this paper we survey the history of this area and point out a number of problems which are still open.
math.GR↗
On the joint spectral radius for isometries of non-positively curved spaces and uniform growth
We recast the notion of joint spectral radius in the setting of groups acting by isometries on non-positively curved spaces and give geometric versions of results of Berger-Wang and Bochi valid for $δ$-hyperbolic spaces and for symmetric spaces of non-compact type. This method produces nice hyperbolic elements in many classical geometric settings. Applications to uniform growth are given, in particular a new proof and a generalization of a theorem of Besson-Courtois-Gallot.
math.GR↗
Diophantine approximation on matrices and Lie groups
We study the general problem of extremality for metric Diophantine approximation on submanifolds of matrices. We formulate a criterion for extremality in terms of a certain family of algebraic obstructions and show that it is sharp. In general, the almost sure diophantine exponent of a submanifold is shown to depend only on its Zariski closure, and when the latter is defined over the rational numbers, we prove that the exponent is rational and give a method to effectively compute it. This method is applied to a number of cases of interest, in particular, we manage to determine the diophantine exponent of random subgroups of certain nilpotent Lie groups in terms of representation theoretic data.
math.NT↗
Random unitaries, amenable linear groups and Jordan's theorem
It is well known that a dense subgroup $G$ of the complex unitary group $U(d)$ cannot be amenable as a discrete group when $d>1$. When $d$ is large enough we give quantitative versions of this phenomenon in connection with certain estimates of random Fourier series on the compact group $\bar G$ that is the closure of $G$. Roughly, we show that if $\bar G$ covers a large enough part of $U(d)$ in the sense of metric entropy then $G$ cannot be amenable. The results are all based on a version of a classical theorem of Jordan that says that if $G$ is finite, or amenable as a discrete group, then $G$ contains an Abelian subgroup with index $e^{o(d^2)}$.
math.RT↗
Salem numbers and the spectrum of hyperbolic surfaces
We give a reformulation of Salem's conjecture about the absence of Salem numbers near one in terms of a uniform spectral gap for certain arithmetic hyperbolic surfaces.
math.NT↗
C*-simplicity and the unique trace property for discrete groups
A discrete group is said to be C*-simple if its reduced C*-algebra is simple, and is said to have the unique trace property if its reduced C*-algebra has a unique tracial state. A dynamical characterization of C*-simplicity was recently obtained by the second and third named authors. In this paper, we introduce new methods for working with group and crossed product C*-algebras that allow us to take the study of C*-simplicity a step further, and in addition to settle the longstanding open problem of characterizing groups with the unique trace property. We give a new and self-contained proof of the aforementioned characterization of C*-simplicity. This yields a new characterization of C*-simplicity in terms of the weak containment of quasi-regular representations. We introduce a convenient algebraic condition that implies C*-simplicity, and show that this condition is satisfied by a vast class of groups, encompassing virtually all previously known examples as well as many new ones. We also settle a question of Skandalis and de la Harpe on the simplicity of reduced crossed products. Finally, we introduce a new property for discrete groups that is closely related to C*-simplicity, and use it to prove a broad generalization of a theorem of Zimmer, originally conjectured by Connes and Sullivan, about amenable actions.
math.OA↗
Lectures on approximate groups and Hilbert's 5th problem
This paper gathers four lectures, based on a mini-course at IMA in 2014, whose aim was to discuss the structure of approximate subgroups of an arbitrary group, following the works of Hrushovski and of Green, Tao and the author. Along the way we discuss the proof of the Gleason-Yamabe theorem on Hilbert's 5th problem about the structure of locally compact groups and explain its relevance to approximate groups. We also present several applications, in particular to uniform diameter bounds for finite groups and to the determination of scaling limits of vertex transitive graphs with large diameter.
math.GR↗
Nilprogressions and groups with moderate growth
We show that doubling at some large scale in a Cayley graph implies uniform doubling at all subsequent scales. The proof is based on the structure theorem for approximate subgroups proved by Green, Tao and the first author. We also give a number of applications to the geometry and spectrum of finite Cayley graphs. For example, we show that a finite group has moderate growth in the sense of Diaconis and Saloff-Coste if and only if its diameter is larger than a fixed power of the cardinality of the group. We call such groups almost flat and show that they have a subgroup of bounded index admitting a cyclic quotient of comparable diameter. We also give bounds on the Cheeger constant, first eigenvalue of the Laplacian, and mixing time. This can be seen as a finite-group version of Gromov's theorem on groups with polynomial growth. It also improves on a result of Lackenby regarding property (tau) in towers of coverings. Another consequence is a universal upper bound on the diameter of all finite simple groups, independent of the CFSG.
math.GR↗
On metric diophantine approximation in matrices and Lie groups
We study the diophantine exponent of analytic submanifolds of the space of m by n real matrices, answering questions of Beresnevich, Kleinbock and Margulis. We identify a family of algebraic obstructions to the extremality of such a submanifold, and give a formula for the exponent when the submanifold is algebraic and defined over the rationals. We then apply these results to the determination of the diophantine exponent of rational nilpotent Lie groups.
math.NT↗
Approximate subgroups and super-strong approximation
Surveying some of the recent developments on approximate subgroups and super-strong approximation for thin groups, we describe the Bourgain-Gamburd method for establishing spectral gaps for finite groups and the proof of the classification of approximate subgroups of semisimple algebraic groups over finite fields. We then give a proof of the super-strong approximation for mod $p$ quotients via random matrix products and a quantitative version of strong approximation. Some applications to the group sieve are also presented.
math.GR↗
Diophantine properties of nilpotent Lie groups
A finitely generated subgroup Γ of a real Lie group G is said to be Diophantine if there is β> 0 such that non-trivial elements in the word ball B_Γ(n) centered at the identity never approach the identity of G closer than |B_Γ (n)|^{-β}. A Lie group G is said to be Diophantine if for every k > 0, a random k-tuple in G generates a Diophantine subgroup. Semi-simple Lie groups are conjectured to be Diophantine but very little is proven in this direction. We give a characterization of Diophantine nilpotent Lie groups in terms of the ideal of laws of their Lie algebra. In particular we show that nilpotent Lie groups of class at most 5, or derived length at most 2, as well as rational nilpotent Lie groups are Diophantine. We also find that there are non Diophantine nilpotent and solvable (non nilpotent) Lie groups.
math.GR↗
Expansion in finite simple groups of Lie type
We show that random Cayley graphs of finite simple (or semisimple) groups of Lie type of fixed rank are expanders. The proofs are based on the Bourgain-Gamburd method and on the main result of our companion paper, establishing strongly dense subgroups in simple algebraic groups.
math.GR↗
Small doubling in groups
Let A be a subset of a group G = (G,.). We will survey the theory of sets A with the property that |A.A| <= K|A|, where A.A = {a_1 a_2 : a_1, a_2 in A}. The case G = (Z,+) is the famous Freiman--Ruzsa theorem.
math.NT↗
Geometry of Locally Compact Groups of Polynomial Growth and Shape of Large Balls
We get asymptotics for the volume of large balls in an arbitrary locally compact group G with polynomial growth. This is done via a study of the geometry of G and a generalization of P. Pansu's thesis. In particular, we show that any such G is weakly commensurable to some simply connected solvable Lie group S, the Lie shadow of G. We also show that large balls in G have an asymptotic shape, i.e. after a suitable renormalization, they converge to a limiting compact set which can be interpreted geometrically. We then discuss the speed of convergence, treat some examples and give an application to ergodic theory. We also answer a question of Burago about left invariant metrics and recover some results of Stoll on the irrationality of growth series of nilpotent groups.
math.GR↗
On the rate of convergence to the asymptotic cone for nilpotent groups and subFinsler geometry
Addressing a question of Gromov, we give a rate in Pansu's theorem about the convergence in Gromov-Hausdorff metric of a finitely generated nilpotent group equipped with a left-invariant word metric scaled by a factor 1/n towards its asymptotic cone. We show that due to the possible presence of abnormal geodesics in the asymptotic cone, this rate cannot be better than n^{1/2} for general non-abelian nilpotent groups. As a corollary we also get an error term of the form vol(B(n))=cn^d + O(n^{d-2/(3r)}) for the volume of Cayley balls of a nilpotent group with nilpotency class r. We also state a number of related conjectural statements.
math.GR↗
A nilpotent Freiman dimension lemma
We prove that a K-approximate subgroup of an arbitrary torsion-free nilpotent group can be covered by a bounded number of cosets of a nilpotent subgroup of bounded rank, where the bounds are explicit and depend only on K. The result can be seen as a nilpotent analogue to Freiman's dimension lemma.
math.CO↗