For a wide range of functions $W\colon\mathbb{N}\to\mathbb{N}$, we establish a general result for estimating weighted averages of the form\[\mathbb{E}^{W}_{n \le N} f(\vartheta(n))= \frac{1}{W(N)}\sum_{n=1}^N (W(n)-W(n-1))f(\vartheta(n)),\]where $f\colon \{1,\ldots,N\}\to\mathbb{C}$ is an arbitrary function, and $\vartheta(n)$ is any arithmetic function that adheres to a certain Gaussian distribution condition. (For instance, one may take $\vartheta(n)=Ω(n)$, where $Ω(n)$ counts the number of prime factors of $n$ with multiplicity, or $\vartheta(n)=s_q(p_n)$, where $s_q$ is the sum-of-digits function in base $q$ and $p_n$ denotes the $n$-th prime. Additional natural examples are discussed in the paper.) Building on our main theorem, we show that if $h(n)$ is a function from a Hardy field with polynomial growth then $(h(\vartheta(n)))_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if one of the following (mutually exclusive) conditions is satisfied: (i) $\lim_{x\to\infty} \frac{|h(x)-p(x)|}{x \log x}=\infty$ for all $p(x)\in \mathbb{Q}[x]$; (ii) $\lim_{x\to\infty}\frac{|h(x)-p(x)|}{\sqrt{x}}=\infty$ for each $p(x)\in \mathbb{Q}[x]$ and there exists $q(x)\in \mathbb{Q}[x]$ such that $\lim_{x\to\infty}\frac{|h(x)-q(x)|}{x}<\infty$. This leads to several novel applications. For example, it follows that $(Ω(n)^c)_{n\in\mathbb{N}}$ is uniformly distributed mod $1$ if and only if $c$ is a non-integer greater than $\frac{1}{2}$.
We use the theory of unipotent flows to prove, in a general setting, the equidistribution of Hecke points. We follow the approach of Burger-Sarnak, and use a theorem of Mozes-Shah.
This is an introduction to the topics of the title, from the 2017 Grenoble Summer school on Arakelov geometry and arithmetic applications. We review Arithmetic intersection numbers, explain the definition of the height of a variety and its properties, both in the framework of classical Arakelov geometry and of Zhang's adelic formalism. We then discuss arithmetic ampleness and its application to the equidistribution theorem of Szpiro-Ullmo-Zhang-Yuan, both at complex and nonarchimedean places. We conclude with Ullmo-Zhang's proof of the Bogomolov conjecture over number fields.
An important ingredient in the Ferrero--Washington proof of the vanishing of cyclotomic $μ$-invariant for Kubota--Leopoldt $p$-adic $L$-functions is an equidistribution result which they established using the Weyl criterion. The purpose of our manuscript is to provide an alternative proof by adopting a dynamical approach. A key ingredient to our methods is studying an ergodic skew-product map on $\mathbb{Z}_p \times [0,1]$, which is then suitably identified as a factor of the $2$-sided Bernoulli shift on the sample space $\{0,1,2,\cdots,p-1\}^{\mathbb{Z}}$.
We give an introduction to adelic mixing and its applications for mathematicians knowing about the mixing of the geodesic flow on hyperbolic surfaces. We focus on the example of the Hecke trees in the modular surface.
We prove distribution estimates for primes in arithmetic progressions to large smooth squarefree moduli, with respect to congruence classes obeying Chinese Remainder Theorem conditions, obtaining an exponent of distribution $\frac{1}{2} + \frac{7}{300}$.
We develop the relation between hyperbolic geometry and arithmetic equidistribution problems that arises from the action of arithmetic groups on real hyperbolic spaces, especially in dimension up to 5. We prove generalisations of Mertens' formula for quadratic imaginary number fields and definite quaternion algebras over the rational numbers, counting results of quadratic irrationals with respect to two different natural complexities, and counting results of representations of (algebraic) integers by binary quadratic, Hermitian and Hamiltonian forms with error bounds. For each such statement, …
We develop the relationship between quaternionic hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on quaternionic hyperbolic spaces, especially in dimension $2$. We prove a Mertens counting formula for the rational points over a definite quaternion algebra $A$ over $\mathbb Q$ in the light cone of quaternionic Hermitian forms, as well as a Neville equidistribution theorem of the set of rational points over $A$ in quaternionic Heisenberg groups.
We strongly develop the relationship between complex hyperbolic geometry and arithmetic counting or equidistribution applications, that arises from the action of arithmetic groups on complex hyperbolic spaces, especially in dimension $2$. We prove a Mertens' formula for the integer points over a quadratic imaginary number fields $K$ in the light cone of Hermitian forms, as well as an equidistribution theorem of the set of rational points over $K$ in Heisenberg groups. We give a counting formula for the cubic points over $K$ in the complex projective plane whose Galois conjugates are orthogonal…
We propose a criterion of equidistribution by the differentiability of certain arithmetic invariants. Combined with the slope method and the asymptotic measures, this criterion gives a new "conceptual" proof to equidistribution results originally obtained via the variation principle.
We prove a Hilbert-Samuel type result of arithmetic big line bundles in Arakelov geometry, which is an analogue of a classical theorem of Siu. An application of this result gives equidistribution of small points over algebraic dynamical systems, following the work of Szpiro-Ullmo-Zhang. We also generalize Chambert-Loir's non-archimedean equidistribution.
Motivated by problems of mathematical physics (quantum chaos) questions of equidistribution of eigenfunctions of the Laplace operator on a Riemannian manifold have been studied by several authors. We consider here, in analogy with arithmetic hyperbolic surfaces, orthonormal bases of eigenfunctions of the Laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of Hecke operators which act on these spherical harmonics. We formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moment…
These are lecture notes from a course in arithmetic dynamics given in Grenoble in June 2017. The main purpose of this text is to explain how arithmetic equidistribution theory can be used in the dynamics of rational maps on P^1. We first briefly introduce the basics of the iteration theory of rational maps on the projective line over C, as well as some elements of iteration theory over an arbitrary complete valued field and the construction of dynamically defined height functions for rational functions defined over \overline Q. The equidistribution of small points gives some original informati…
Baker-Rumely and Favre-Rivera-Letelier independently proved an important arithmetic equidistribution theorem for points of small height on the Berkovich compactification of the projective line with respect to an adelic measure on $\mathbb{P}^1$. Around the same time, Chambert-Loir proved a more general version of this arithmetic equidistribution theorem in the setting of curves from a different approach. We generalize the notion of an adelic measure to that of a quasi-adelic measure on $\mathbb{P}^1$, and show that arithmetic equidistribution of points with small height holds for quasi-adelic …
This paper is devoted to the study of equidistributional properties of \textit{totient points} in $\mathbb{N}^r$, that is, of coprime $r$-tuples of integers, with particular emphasis on some relevant sets of totient points fulfilling extra divisibility or coprimality conditions, or lying on arithmetic progressions.
We study the distribution, in the space of Satake parameters, of local components of Siegel cusp forms of genus 2 and growing weight, subject to a specific weighting which allows us to apply results concerning Bessel models and a variant of Petersson's formula. We obtain for this family a quantitative local equidistribution result, and derive a number of consequences. In particular, we show that the computation of the density of low-lying zeros of the spinor L-functions (for restricted test functions) gives global evidence for a well-known conjecture of Böcherer concerning the arithmetic natur…
Keating and Rudnick studied the variance of the polynomial von Mangoldt function $Λ\colon \mathbb{F}_q[t] \rightarrow \mathbb{C}$ in arithmetic progressions and short intervals using two equidistribution results by Katz. Hall, Keating and Roditty-Gershon then generalised the result for arithmetic progressions for a von Mangoldt function $Λ_ρ$ attached to a Galois representation $ρ\colon \mathrm{Gal} ( \overline{\mathbb{F}_q(t)}/\mathbb{F}_q(t) ) \rightarrow \mathrm{GL}_m(\overline{\mathbb{Q}}_\ell)$. We employ a recent equidistribution result by Sawin in order to generalise the corresponding r…
We study the variance of sums of the indicator function of square-full polynomials in both arithmetic progressions and short intervals. Our work is in the context of the ring $F_{q}[T]$ of polynomials over a finite field $F_{q}$ of $q$ elements, in the limit $q\rightarrow\infty$. We use a recent equidistribution result due to N. Katz to express these variances in terms of triple matrix integrals over the unitary group, and evaluate them.
We study a refinement of the quantum unique ergodicity conjecture for shrinking balls on arithmetic hyperbolic manifolds, with a focus on dimensions $ 2 $ and $ 3 $. For the Eisenstein series for the modular surface $\mathrm{PSL}_2( {\mathbb Z}) \backslash \mathbb{H}^2$ we prove failure of quantum unique ergodicity close to the Planck-scale and an improved bound for its quantum variance. For arithmetic $ 3 $-manifolds we show that quantum unique ergodicity of Hecke-Maaß forms fails on shrinking balls centered on an arithmetic point and radius $ R \asymp t_j^{-δ} $ with $ δ> 3/4 $. For $ \mathr…
Let $f=\sum_{n=1}^{\infty}a(n)q^{n}\in S_{k+1/2}(N,χ_{0})$ be a non-zero cuspidal Hecke eigenform of weight $k+\frac{1}{2}$ and the trivial nebentypus $χ_{0}$ where the Fourier coefficients $a(n)$ are real. Bruinier and Kohnen conjectured that the signs of $a(n)$ are equidistributed. This conjecture was proved to be true by Inam, Wiese and Arias-de-Reyna for the subfamilies $\{a(t n^{2})\}_{n}$ where $t$ is a squarefree integer such that $a(t)\neq 0$. Let $q$ and $d$ be natural numbers such that $(d,q)=1$. In this work, we show that $\{a(t n^{2})\}_{n}$ is equidistributed over any arithmetic p…
We prove effective equidistribution theorems, with polynomial error rate, for orbits of the unipotent subgroups of $\operatorname{SL}_2(\mathbb R)$ in arithmetic quotients of $\operatorname{SL}_2(\mathbb C)$ and $\operatorname{SL}_2(\mathbb R)\times\operatorname{SL}_2(\mathbb R)$. The proof is based on the use of a Margulis function, tools from incidence geometry, and the spectral gap of the ambient space.
We compute the variances of sums in arithmetic progressions of arithmetic functions associated with certain $L$-functions of degree two and higher in $\mathbb{F}_q[t]$, in the limit as $q\to\infty$. This is achieved by establishing appropriate equidistribution results for the associated Frobenius conjugacy classes. The variances are thus related to matrix integrals, which may be evaluated. Our results differ significantly from those that hold in the case of degree-one $L$-functions (i.e. situations considered previously using this approach). They correspond to expressions found recently in the…
The densities of small linear structures (such as arithmetic progressions) in subsets of Abelian groups can be expressed as certain analytic averages involving linear forms. Higher-order Fourier analysis examines such averages by approximating the indicator function of a subset by a function of bounded number of polynomials. Then, to approximate the average, it suffices to know the joint distribution of the polynomials applied to the linear forms. We prove a near-equidistribution theorem that describes these distributions for the group $\mathbb{F}_p^n$ when $p$ is a fixed prime. This fundament…
We prove that arithmetic quantum unique ergodicity holds on compact arithmetic quotients of $GL(2,\mathbb{Q}_p)$ for automorphic forms belonging to the principal series. We interpret this conclusion in terms of the equidistribution of eigenfunctions on covers of a fixed regular graph or along nested sequences of regular graphs. Our results are the first of their kind on any p-adic arithmetic quotient. They may be understood as analogues of Lindenstrauss's theorem on the equidistribution of Maass forms on a compact arithmetic surface. The new ingredients here include the introduction of a repre…
We consider Bourgain's ergodic theorem regarding arithmetic averages in the cases where quantitative mixing is present in the dynamical system. Focusing on the case of the horocyclic flow, those estimates allows us to bound from above the Hausdorff dimension of the exceptional set, providing evidence towards conjectures by Margulis,Shah and Sarnak regarding equidistribution of arithmetic averages in homogeneous spaces. We also prove the existence of a uniform upper bound for the Hausdorff dimension of the exceptional set which is independent from the spectral gap.
We calculate the mean and variance of sums of the Möbius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, whi…
We show that the multiple divisor functions of integers in invertible residue classes modulo a prime number, as well as the Fourier coefficients of GL(N) Maass cusp forms for all N larger than 2, satisfy a central limit theorem in a suitable range, generalizing the case N=2 treated by E. Fouvry, S. Ganguly, E. Kowalski and P. Michel. Such universal Gaussian behaviour relies on a deep equidistribution result of products of hyper-Kloosterman sums.
We establish results with an arithmetic flavor that generalize the polynomial multidimensional Szemeredi theorem and related multiple recurrence and convergence results in ergodic theory. For instance, we show that in all these statements we can restrict the implicit parameter to those integers that have an even number of distinct prime factors, or satisfy any other congruence condition. In order to obtain these refinements we study the limiting behavior of some closely related multiple ergodic averages with weights given by appropriately chosen multiplicative functions. These averages are the…
We give an ergodic theoretic proof of a theorem of Duke about equidistribution of closed geodesics on the modular surface. The proof is closely related to the work of Yu. Linnik and B. Skubenko, who in particular proved this equidistribution under an additional congruence assumption on the discriminant. We give a more conceptual treatment using entropy theory, and show how to use positivity of the discriminant as a substitute for Linnik's congruence condition.
The Chowla conjecture asserts that the values of the Liouville function form a normal sequence of plus and minus ones. Reinterpreted in the language of ergodic theory it asserts that the Liouville function is generic for the Bernoulli measure on the space of sequences with values plus or minus one. We show that these statements are implied by the much weaker hypothesis that the Liouville function is generic for an ergodic measure. We also give variants of this result related to a conjecture of Elliott on correlations of multiplicative functions with values on the unit circle. Our argument has …
Let $f$ be a rational function of degree $d>1$ on the projective line over a possibly non-archimedean algebraically closed field. A well-known process initiated by Brolin considers the pullbacks of points under iterates of $f$, and produces an important equilibrium measure. We define the asymptotic Fekete property of pullbacks of points, which means that they mirror the equilibrium measure appropriately. As application, we obtain an error estimate of equidistribution of pullbacks of points for $C^1$-test functions in terms of the proximity of wandering critical orbits to the initial points, an…
For a projective variety X defined over a field K, there is a special class of self-morphisms of X called algebraic dynamical systems. In this paper we take K to be the function field of a smooth curve and prove that at each place of K, subvarieties of X of dynamically small height are equidistributed on the associated Berkovich analytic space. We carefully develop all of the arithmetic intersection theory needed to state and prove this theorem, and we present several applications on the non-Zariski density of preperiodic points and of points of small height in field extensions of bounded degr…
Let $μ$ be a probability measure on $\mathbb{R}/\mathbb{Z}$ that is ergodic under the $\times p$ map, with positive entropy. In 1995, Host showed that if $\gcd(m,p)=1$ then $μ$ almost every point is normal in base $m$. In 2001, Lindenstrauss showed that the conclusion holds under the weaker assumption that $p$ does not divide any power of $m$. In 2015, Hochman and Shmerkin showed that this holds in the "correct" generality, i.e. if $m$ and $p$ are independent. We prove a simultaneous version of this result: for $μ$ typical $x$, if $m>p$ are independent, we show that the orbit of $(x,x)$ under …
The main goal of this work is to establish quantitative nondivergence estimates for flows on homogeneous spaces of products of real and $p$-adic Lie groups. These results have applications both to ergodic theory and to Diophantine approximation. Namely, earlier results of Dani (finiteness of locally finite ergodic unipotent-invariant measures on real homogeneous spaces) and Kleinbock-Margulis (strong extremality of nondegenerate submanifolds of $\Bbb R^n$) are generalized to the $S$-arithmetic setting.
We establish mean convergence for multiple ergodic averages with iterates given by distinct fractional powers of primes and related multiple recurrence results. A consequence of our main result is that every set of integers with positive upper density contains patterns of the form $\{m,m+[p_n^a], m+[p_n^b]\}$, where $a,b$ are positive non-integers and $p_n$ denotes the $n$-th prime, a property that fails if $a$ or $b$ is a natural number. Our approach is based on a recent criterion for joint ergodicity of collections of sequences and the bulk of the proof is devoted to obtaining good seminorm …
