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Amos Nevo

Publications and source records attributed to Amos Nevo.

At least 19 recordsLinked to original sources
arXiv2025

Hardy-Littlewood maximal operator on spaces of exponential volume growth

We consider the Hardy-Littlewood maximal function associated with ball averages on spaces with exponential volume growth. We focus on discrete groups with balls defined by invariant metrics associated with a variety of length functions. Under natural assumptions on the rough radial structure of the group in question, we establish a weak-type $\mathcal{L}\left(\log \mathcal{L}\right)^{\bf c}$ maximal inequality for the Hardy-Littlewood maximal function. We give a variety of examples where the rough radial structure assumptions hold, based on considerations from geometric group theory, or on analytic considerations related to the regular representation of the group. We elucidate the connections of these assumptions to a spherical coarse median inequality, to almost exact polynomial-exponential growth of balls, and to the radial rapid decay property. In particular, the weak-type maximal inequality in $\mathcal{L}\left(\log \mathcal{L}\right)^{\bf c}$ is established for any lattice in a connected semisimple Lie group with finite center, with respect to the distance function restricted from the Riemannian distance on symmetric space to an orbit of the lattice. It is also established for right-angled Artin groups, Coxeter groups and braid groups, for a suitable choice of word metric. For non-elementary word-hyperbolic group we establish that the Hardy-Littlewood maximal operator with respect to balls defined by a word length satisfies the weak-type $(1,1)$ maximal inequality, which is the optimal result.

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arXiv2024

Automorphic density estimates and optimal Diophantine exponents

The present paper is devoted to establishing an optimal approximation exponent for the action of an irreducible uniform lattice subgroup of a product group on its proper factors. Previously optimal approximation exponents for lattice actions on homogeneous spaces were established under the assumption that the restriction of the automorphic representation to the stability group is suitably tempered. However, for irreducible lattices in semisimple algebraic groups, either this property does not hold or it amounts to an instance of the Ramanujan-Petersson-Selberg conjecture. Sarnak's Density Hypothesis and its variants bounding the multiplicities of irreducible representations occurring in the decomposition of the automorphic representation can be viewed as a weakening of the temperedness property. A refined form of this hypothesis has recently been established for uniform irreducible arithmetic congruence lattices arising from quaternion algebras. We employ this result in order to establish - unconditionally - an optimal approximation exponent for the actions of these lattices on the associated symmetric spaces. We also give a general spectral criterion for the optimality of the approximation exponent for irreducible uniform lattices in a product of arbitrary Gelfand pairs. Our methods involve utilizing the multiplicity bounds in the pre-trace formula, establishing refined estimates of the spherical transforms, and carrying out an elaborate spectral analysis that bounds the Hilbert-Schmidt norms of carefully balanced geometric convolution operators.

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arXiv2023

Shannon-McMillan-Breiman theorem along almost geodesics in negatively curved groups

Consider a non-elementary Gromov-hyperbolic group $Γ$ with a suitable invariant hyperbolic metric, and an ergodic probability measure preserving (p.m.p.) action on $(X,μ)$. We construct special increasing sequences of finite subsets $F_n(y)\subset Γ$, with $(Y,ν)$ a suitable probability space, with the following properties: given any countable partition $\mathcal{P}$ of $X$ of finite Shannon entropy, the refined partitions $\bigvee_{γ\in F_n(y)}γ\mathcal{P}$ have normalized information functions which converge to a constant limit, for $μ$-almost every $x\in X$ and $ν$-almost every $y\in Y$; the sets $\mathcal{F}_n(y)$ constitute almost-geodesic segments, and $\bigcup_{n\in \mathbb{N}} F_n(y)$ is a one-sided almost geodesic with limit point $F^+(y)\in \partial Γ$, starting at a fixed bounded distance from the identity, for almost every $y\in Y$; the distribution of the limit point $F^+(y)$ belongs to the Patterson-Sullivan measure class on $\partial Γ$ associated with the invariant hyperbolic metric. The main result of the present paper amounts therefore to a Shannon-McMillan-Breiman theorem along almost geodesic segments in any p.m.p. action of $Γ$ as above. For several important classes of examples we analyze, the construction of $F_n(y)$ is purely geometric and explicit. Furthermore, consider the infimum of the limits of the normalized information functions, taken over all $Γ$-generating partitions of $X$. Using an important inequality due to B. Seward, we deduce that it is equal to the Rokhlin entropy $\frak{h}^{\text{Rok}}$ of the $Γ$-action on $(X,μ)$, provided that the action is free.

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arXiv2022

On discrepancy, intrinsic Diophantine approximation, and spectral gaps

In the present paper we establish bounds for the size of the spectral gap for actions of algebraic groups on certain homogeneous spaces. Our approach is based on estimating operator norms of suitable averaging operators, and we develop techniques for establishing both upper and lower bounds for such norms. We shall show that this analytic problem is closely related to the arithmetic problem of establishing bounds on the discrepancy of distribution for rational points on algebraic group varieties. As an application, we show how to establish an effective bound for property $τ$ of congruence subgroups of arithmetic lattices in algebraic groups which are forms of $SL(2)$, using estimates in intrinsic Diophantine approximation which follow from Heath-Brown's analysis of rational points on 3-dimensional quadratic surfaces.

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arXiv2021

Discrepancy of rational points in simple algebraic groups

The present paper analyzes the discrepancy of distribution of rational points on general semisimple algebraic group varieties. The results include mean-square, almost sure, and uniform discrepancy estimates with explicit error bounds, which apply to general families of subsets, and are valid at arbitrarily small scales. We also consider an analogue of W. Schmidt's classical theorem, which establishes effective almost sure asymptotic counting of rational solutions to Diophantine inequalities in Euclidean spaces. We formulate and prove a version of it for rational points on the group variety, together with an effective bound which in some instances can be expected to be best possible.

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arXiv2019

Effective counting on translation surfaces

We prove an effective version of a celebrated result of Eskin and Masur: for any affine invariant manifold of translation surfaces, almost every translation surface has quadratic growth for the saddle connection holonomy vectors, with an effective bound of the error. We also provide effective versions of counting in sectors and in ellipses.

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arXiv2019

Effective counting for discrete lattice orbits in the plane via Eisenstein series

We prove effective bounds on the rate in the quadratic growth asymptotics for the orbit of a non-uniform lattice of SL(2,R), acting linearly on the plane. This gives an error bound in the count of saddle connection holonomies, for some Veech surfaces. The proof uses Eisenstein series and relies on earlier work of many authors (notably Selberg). Our results improve earlier error bounds for counting in sectors and in smooth star shaped domains.

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arXiv2019

The Shannon-McMillan-Breiman theorem beyond amenable groups

We introduce a new isomorphism-invariant notion of entropy for measure preserving actions of arbitrary countable groups on probability spaces, which we call orbital Rokhlin entropy. It employs Danilenko's orbital approach to entropy of a partition, and is motivated by Seward's recent generalization of Rokhlin entropy from amenable to general groups. A key ingredient in our approach is the use of an auxiliary probability-measure-preserving hyperfinite equivalence relation. Under the assumption of ergodicity of the auxiliary equivalence relation, our main result is a Shannon-McMillan-Breiman pointwise almost sure convergence theorem for the orbital entropy of partitions in measure-preserving group actions, the first such convergence result going beyond the realm of amenable groups. As a special case, we obtain a Shannon-McMillan-Breiman theorem for all strongly mixing actions of any countable group. Furthermore, we compare orbital Rokhlin entropy to Rokhlin entropy, and using an important recent result of Seward we show that they coincide for free, ergodic actions of any countable group. Finally, we consider actions of non-Abelian free groups and demonstrate the geometric significance of the entropy equipartition property implied by the Shannon-McMillan-Breiman theorem. We show that the orbital entropy of a partition is the limit of the information functions of the sequence of partitions arising from refining any given finite partition along almost every horoball in the group.

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arXiv2018

Optimal density for values of generic polynomial maps

We establish that the optimal bound for the size of the smallest integral solution of the Oppenheim Diophantine approximation problem $\abs{Q(x)-ξ}< ε$ for a generic ternary form $Q$ is $\abs{x}\ll ε^{-1}$. We also establish an optimal rate of density for the values of polynomials maps in a number of other natural problems, including the values of linear forms restricted to suitable quadratic surfaces, and the values of the polynomial map defined by the generators of the ring of conjugation-invariant polynomials on $M_3(\C)$. These results are instances of a general approach that we develop, which considers a rational affine algebraic subvariety of Euclidean space, invariant and homogeneous under an action of a semisimple Lie group $G$. Given a polynomial map $F$ defined on the Euclidean space which is invariant under a semisimple subgroup $H$ of the acting group $G$, consider the family of its translates $F\circ g$ by elements of the group. We study the restriction of these polynomial functions to the integer points on the variety confined to a large Euclidean ball. Our main results establish an explicit rate of density for their values, for generic polynomials in the family. This problem has been extensively studied before when the polynomials in question are linear, in the context of classical Diophantine approximation, but very little was known about it for polynomial of higher degree. We formulate a heuristic pigeonhole lower bound for the density and an explicit upper bound for it, formulate a sufficient condition for the coincidence of the lower and upper bounds, and in a number of natural examples establish that they indeed match. Finally, we also establish a rate of density for values of homogeneous polynomials on homogeneous projective varieties.

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arXiv2017

Hyperbolic geometry and pointwise ergodic theorems

We establish pointwise ergodic theorems for a large class of natural averages on simple Lie groups of real-rank-one, going well beyond the radial case considered previously. The proof is based on a new approach to pointwise ergodic theorems, which is independent of spectral theory. Instead, the main new ingredient is the use of direct geometric arguments in hyperbolic space.

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arXiv2017

Prime Points in Orbits: Some Instances of the Bourgain-Gamburd-Sarnak Conjecture

We use Vaughan's variation on Vinogradov's three-primes theorem to prove Zariski-density of prime points in several infinite families of hypersurfaces, including level sets of some quadratic forms, the Permanent polynomial, and the defining polynomials of some pre-homogeneous vector spaces. Three of these families are instances of a conjecture by Bourgain, Gamburd and Sarnak regarding prime points in orbits of simple algebraic groups. Our approach is based on the formulation of a general condition on the defining polynomial of a hypersurface, which suffices to guarantee that Zariski-density of prime points is equivalent to the existence of an odd point.

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arXiv2016

Horospherical coordinates of lattice points in hyperbolic space: effective counting and equidistribution

We establish effective counting and equidistribution results for lattice points in families of domains in hyperbolic spaces, of any dimension and over any field. The domains we focus on are defined as product sets with respect to the Iwasawa decomposition. Several classical Diophantine problems can be reduced to counting lattice points in such domains, including distribution of shortest solution to the gcd equation, and angular distribution of primitive vectors in the plane. We give an explicit and effective solution to these problems, and extend them to imaginary quadratic number fields. Further applications include counting lifts of closed horospheres to hyperbolic manifolds and establishing an equidistribution property of integral solutions to the Diophantine equation defined by a Lorentz form.

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arXiv2016

Counting lattice points in norm balls on higher rank simple Lie groups

We establish an error estimate for counting lattice points in Euclidean norm balls (associated to an arbitrary irreducible linear representation) for lattices in simple Lie groups of real rank at least two. Our approach utilizes refined spectral estimates based on the existence of universal pointwise bounds for spherical functions on the groups involved. We focus particularly on the case of the special linear groups where we give a detailed proof of error estimates which constitute the first improvement of the best current bound established by Duke, Rudnick and Sarnak in 1991, and are nearly twice as good in some cases.

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arXiv2014

Best possible rates of distribution of dense lattice orbits in homogeneous spaces

The present paper establishes upper and lower bounds on the speed of approximation in a wide range of natural Diophantine approximation problems. The upper and lower bounds coincide in many cases, giving rise to optimal results in Diophantine approximation which were inaccessible previously. Our approach proceeds by establishing, more generally, upper and lower bounds for the rate of distribution of dense orbits of a lattice subgroup $Γ$ in a connected Lie (or algebraic) group $G$, acting on suitable homogeneous spaces $G/H$. The upper bound is derived using a quantitative duality principle for homogeneous spaces, reducing it to a rate of convergence in the mean ergodic theorem for a family of averaging operators supported on $H$ and acting on $G/Γ$. In particular, the quality of the upper bound on the rate of distribution we obtain is determined explicitly by the spectrum of $H$ in the automorphic representation on $L^2(Γ\setminus G)$. We show that the rate is best possible when the representation in question is tempered, and show that the latter condition holds in a wide range of examples.

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arXiv2014

The lattice point counting problem on the Heisenberg groups

We consider the radial and Heisenberg-homogeneous norms on the Heisenberg groups given by $N_{α,A}((z,t)) = \left(|z|^α+ A |t|^{α/2}\right)^{1/α}$, for $α\ge 2$ and $A>0$. This natural family includes the canonical Cygan-Korányi norm, corresponding to $α=4$. We study the lattice points counting problem on the Heisenberg groups, namely establish an error estimate for the number of points that the lattice of integral points has in a ball of large radius $R$. The exponent we establish for the error in the case $α=2$ is the best possible, in all dimensions.

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arXiv2014

Diophantine approximation exponents on homogeneous varieties

Recent years have seen very important developments at the interface of Diophantine approximation and homogeneous dynamics. In the first part of the paper we give a brief exposition of a dictionary developed by Dani and Kleinbock-Margulis which relates Diophantine properties of vectors to distribution of orbits of flows on the space of unimodular lattices. In the second part of the paper we briefly describe an extension of this dictionary recently developed by the authors, which establishes an analogous dynamical correspondence for general lattice orbits on homogeneous spaces. We concentrate specifically on the problem of estimating exponents of Diophantine approximation by arithmetic lattices acting on algebraic varieties. In the third part of the paper, we exemplify our results by establishing explicit bounds for the Diophantine exponent of dense lattice orbits in a number of basic cases. These include the linear and affine actions on affine spaces, and the action on the variety of matrices of fixed determinant. In some cases, these exponents are shown to be best possible.

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