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## Equidistribution and variation of height functions

Sukiennik, Justin E. (1981 - ); Tucker, Thomas J.
Fonte: University of Rochester Publicador: University of Rochester
Tipo: Tese de Doutorado Formato: Number of Pages:vii, 51 leaves
ENG
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Thesis (Ph. D.)--University of Rochester. Dept. of Mathematics, 2009.; We investigate an assortment of questions about variations of heights in a number field K and a function field Fp(T), and the effect these variations have on the distribution of points. A theorem of Y. Bilu describes the equidistribution of points (and their conjugates) with small heights around the unit circle in P1(ℚ); we first examine the conditions to which this theorem can be strengthened by giving a generalized version of a counterexample done by P. Autissier. Next, we find a strict upper bound for the differences of the height of a point and its affine linear image, and conclude there is an asymmetry when we interchange the difference terms. This seems counterintuitive in light of a result by C. Petsche, L. Szpiro, and T. Tucker. Lastly, we prove an S-integrality theorem that answers similar questions to those raised by a conjecture of S. Ih. More specically, for z ∈ Fp(T) non preperiodic for certain polynomials f ∈ Fp(T), we demonstrate that only finitely many mth iterates fm(z) are S-integral with respect to 0.

## Small scale equidistribution of eigenfunctions on the torus

Lester, Stephen; Rudnick, Zeév
Tipo: Artigo de Revista Científica
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We study the small scale distribution of the $L^2$ mass of eigenfunctions of the Laplacian on the flat torus $\mathbb T^d$. Given an orthonormal basis of eigenfunctions, we show the existence of a density one subsequence whose $L^2$ mass equidistributes at small scales. In dimension two our result holds all the way down to the Planck scale. For dimensions $d=3,4$ we can restrict to individual eigenspaces and show small scale equidistribution in that context. We also study irregularities of quantum equidistribution: We construct eigenfunctions whose $L^2$ mass does not equidistribute at all scales above the Planck scale. Additionally, in dimension $d=4$ we show the existence of eigenfunctions for which the proportion of $L^2$ mass in small balls blows up at certain scales.

## Equidistribution, ergodicity and irreducibility in CAT(-1) spaces

Tipo: Artigo de Revista Científica
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We prove an equidistribution theorem a la Bader-Muchnik for operator-valued measures associated with boundary representations in the context of discrete groups of isometries of CAT(-1) spaces thanks to an equidistribution theorem of T. Roblin. This result can be viewed as a generalization of Birkhoff's ergodic theorem for quasi invariant measures. In particular, this approach gives a dynamical proof of the fact that boundary representations are irreducible. Moreover, we prove some equidistribution results for conformal densities using elementary techniques from harmonic analysis.

## Equidistribution estimates for Fekete points on complex manifolds

Lev, Nir; Ortega-Cerdà, Joaquim
Tipo: Artigo de Revista Científica
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We study the equidistribution of Fekete points in a compact complex manifold. These are extremal point configurations defined through sections of powers of a positive line bundle. Their equidistribution is a known result. The novelty of our approach is that we relate them to the problem of sampling and interpolation on line bundles, which allows us to estimate the equidistribution of the Fekete points quantitatively. In particular we estimate the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure. The sampling and interpolation arrays on line bundles are a subject of independent interest, and we provide necessary density conditions through the classical approach of Landau, that in this context measures the local dimension of the space of sections of the line bundle. We obtain a complete geometric characterization of sampling and interpolation arrays in the case of compact manifolds of dimension one, and we prove that there are no arrays of both sampling and interpolation in the more general setting of semipositive line bundles.; Comment: Improved version with a sharp decay rate in the estimate of the Kantorovich-Wasserstein distance of the Fekete points to its limiting measure (Theorem 2)

## On the uniform equidistribution of closed horospheres in hyperbolic manifolds

Södergren, Anders
Tipo: Artigo de Revista Científica
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We prove asymptotic equidistribution results for pieces of large closed horospheres in cofinite hyperbolic manifolds of arbitrary dimension. This extends earlier results by Hejhal and Str\"ombergsson in dimension 2. Our proofs use spectral methods, and lead to precise estimates on the rate of convergence to equidistribution.; Comment: 58 pages

## On Arnold's and Kazhdan's equidistribution problems

Gorodnik, Alexander; Nevo, Amos
Tipo: Artigo de Revista Científica
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We consider isometric actions of lattices in semisimple algebraic groups on (possibly non-compact) homogeneous spaces with (possibly infinite) invariant Radon measure. We assume that the action has a dense orbit, and demonstrate two novel and non-classical dynamical phenomena that arise in this context. The first is the existence of a mean ergodic theorem even when the invariant measure is infinite, which implies the existence of an associated limiting distribution, possibly different than the invariant measure. The second is uniform quantitative equidistribution of all orbits in the space, which follows from a quantitative mean ergodic theorem for such actions. In turn, these results imply quantitative ratio ergodic theorems for isometric actions of lattices. This sheds some unexpected light on certain equidistribution problems posed by Arnol'd and also on the equidistribution conjecture for dense subgroups of isometries formulated by Kazhdan. We briefly describe the general problem regarding ergodic theorems for actions of lattices on homogeneous spaces and its solution via the duality principle \cite{GN2}, and give a number of examples to demonstrate our results. Finally, we also prove results on quantitative equidistribution for transitive actions.; Comment: Submitted

## Equidistribution of points via energy

Pritsker, Igor E.
Tipo: Artigo de Revista Científica
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We study the asymptotic equidistribution of points with discrete energy close to Robin's constant of a compact set in the plane. Our main tools are the energy estimates from potential theory. We also consider the quantitative aspects of this equidistribution. Applications include estimates of growth for the Fekete and Leja polynomials associated with large classes of compact sets, convergence rates of the discrete energy approximations to Robin's constant, and problems on the means of zeros of polynomials with integer coefficients.; Comment: arXiv admin note: text overlap with arXiv:1307.5841

## Pointwise equidistribution with an error rate and with respect to unbounded functions

Kleinbock, Dmitry; Shi, Ronggang; Weiss, Barak
Tipo: Artigo de Revista Científica
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Consider $G=\SL_{ d }(\mathbb R)$ and $\Gamma=\SL_{ d }(\mathbb Z)$. It was recently shown by the second-named author \cite{s} that for some diagonal subgroups $\{g_t\}\subset G$ and unipotent subgroups $U\subset G$, $g_t$-trajectories of almost all points on all $U$-orbits on $G/\Gamma$ are equidistributed with respect to continuous compactly supported functions $\varphi$ on $G/\Gamma$. In this paper we strengthen this result in two directions: by exhibiting an error rate of equidistribution when $\varphi$ is smooth and compactly supported, and by proving equidistribution with respect to certain unbounded functions, namely Siegel transforms of Riemann integrable functions on $\R^d$. For the first part we use a method based on effective double equidistribution of $g_t$-translates of $U$-orbits, which generalizes the main result of \cite{km12}. The second part is based on Schmidt's results on counting of lattice points. Number-theoretic consequences involving spiraling of lattice approximations, extending recent work of Athreya, Ghosh and Tseng \cite{agt1}, are derived using the equidistribution result.; Comment: minor compilation issue fixed

## Quantitative equidistribution properties of toral eigenfunctions

Hezari, Hamid; Riviere, Gabriel
Tipo: Artigo de Revista Científica
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We prove quantitative equidistribution properties for orthonormal bases of eigenfunctions of the Laplacian on the rational $d$-torus. We show that the rate of equidistribution of such eigenfunctions is of polynomial decay. We also prove that equidistribution of eigenfunctions holds for symbols supported in balls with a radius shrinking at a polynomial rate.; Comment: This article is based on the appendix of our previous preprint: arXiv:1411.4078. We have included improvements and have simplified the proofs (no semiclassical/microlocal techniques are necessary)

## Criteria for equidistribution of solutions of word equations on SL(2)

Bandman, Tatiana; Kunyavskii, Boris
Tipo: Artigo de Revista Científica
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We study equidistribution of solutions of word equations of the form w(x,y)=g in the family of finite groups SL(2,q). We provide criteria for equidistribution in terms of the trace polynomial of w. This allows us to get an explicit description of certain classes of words possessing the equidistribution property and show that this property is generic within these classes.; Comment: 21 pages

## Effective equidistribution of twisted horocycle flows and horocycle maps

Flaminio, Livio; Forni, Giovanni; Tanis, James
Tipo: Artigo de Revista Científica
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We prove bounds for twisted ergodic averages for horocycle flows of hyperbolic surfaces, both in the compact and in the non-compact finite area case. From these bounds we derive effective equidistribution results for horocycle maps. As an application of our main theorems in the compact case we further improve on a result of A. Venkatesh, recently already improved by J. Tanis and P. Vishe, on a sparse equidistribution problem for classical horocycle flows proposed by N. Shah and G. Margulis, and in the general non-compact, finite area case we prove bounds on Fourier coefficients of cups forms which are off the best known bounds of A. Good only by a logarithmic term. Our approach is based on Sobolev estimates for solutions of the cohomological equation and on scaling of invariant distributions for twisted horocycle flows.; Comment: 83 pages

## Adelic equidistribution, characterization of equidistribution, and a general equidistribution theorem in non-archimedean dynamics

Okuyama, Yûsuke
Tipo: Artigo de Revista Científica
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We determine when the equidistribution property for possibly moving targets holds for a rational function of degree more than one on the projective line over an algebraically closed field of any characteristic and complete with respect to a non-trivial absolute value. This characterization could be useful in the positive characteristic case. Based on the variational argument, we give a purely local proof of the adelic equidistribution theorem for possibly moving targets, which is due to Favre and Rivera-Letelier, using a dynamical Diophantine approximation theorem by Silverman and by Szpiro--Tucker. We also give a proof of a general equidistribution theorem for possibly moving targets, which is due to Lyubich in the archimedean case and due to Favre and Rivera-Letelier for constant targets in the non-archimedean and any characteristic case and for moving targets in the non-archimedean and 0 characteristic case.; Comment: 25 pages, no figures. (v2: a few minor modifications)

## Equidistribution of Zeros of Random Holomorphic Sections for Moderate Measures

Shao, Guokuan
Tipo: Artigo de Revista Científica
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We establish an equidistribution theorem for the zeros of random holomorphic sections of high powers of a positive holomorphic line bundle. The equidistribution is associated with a family of singular moderate measures. We also give a convergence speed for the equidistribution.; Comment: 18 pages

## Effective equidistribution for some unipotent flows in PSL(2, R)^k mod cocompact, irreducible lattice

Tanis, James
Tipo: Artigo de Revista Científica
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Let $k \geq 2$, and let $\Gamma \subset \operatorname{PSL}(2, \mathbb{R})^k$ be an irreducible, cocompact lattice. We prove effective equidistribution for coordinate horocycle flows on $\Gamma \backslash \operatorname{PSL}(2, \mathbb{R})^k$. This is the simplest case for proving effective equidistribution of unipotent flows in this setting. The main ingredients are Flaminio-Forni's study of the equidistribution of the horocycle flow and a result by Kelmer-Sarnak on the strong spectral gap property of $\Gamma$ in $\operatorname{PSL}(2, \mathbb{R})^k$.; Comment: 12 pages, minor changes

## Sparse equidistribution problems, period bounds, and subconvexity

Venkatesh, Akshay
Tipo: Artigo de Revista Científica
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We introduce a geometric'' method to bound periods of automorphic forms. The key features of this method are the use of equidistribution results in place of mean value theorems, and the systematic use of mixing and the spectral gap. Applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of CM points; to subconvexity of the triple product period in the level aspect over number fields, which implies subconvexity for certain standard and Rankin-Selberg $L$-functions; and to bounding Fourier coefficients of automorphic forms.; Comment: Minor revisions made

## Counting and equidistribution in Heisenberg groups

Parkkonen, Jouni; Paulin, Frédéric
Tipo: Artigo de Revista Científica
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27.36%
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 and isotropic for a given Hermitian form over $K$, and a counting and equidistribution result for arithmetic chains in the Heisenberg group when their Cygan diameter tends to $0$.; Comment: 35 pages. Lemma 8 clarifies some results, including Theorem 1

## Equidistribution over function fields

Gubler, Walter
Tipo: Artigo de Revista Científica
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We prove equidistribution of a generic net of small points in a projective variety X over a function field K. For an algebraic dynamical system over K, we generalize this equidistribution theorem to a small generic net of subvarieties. For number fields, these results were proved by Yuan and we transfer here his methods to function fields. If X is a closed subvariety of an abelian variety, then we can describe the equidistribution measure explicitly in terms of convex geometry.; Comment: 23 pages; reference to X.W.C. Faber added who obtained some of the results independently. Minor errors corrected. To appear in manuscripta mathematica

## Equidistribution of Kronecker sequences along closed horocycles

Marklof, Jens; Strombergsson, Andreas
Tipo: Artigo de Revista Científica
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27.36%
It is well known that (i) for every irrational number $\alpha$ the Kronecker sequence $m\alpha$ ($m=1,...,M$) is equidistributed modulo one in the limit $M\to\infty$, and (ii) closed horocycles of length $\ell$ become equidistributed in the unit tangent bundle $T_1 M$ of a hyperbolic surface $M$ of finite area, as $\ell\to\infty$. In the present paper both equidistribution problems are studied simultaneously: we prove that for any constant $\nu > 0$ the Kronecker sequence embedded in $T_1 M$ along a long closed horocycle becomes equidistributed in $T_1 M$ for almost all $\alpha$, provided that $\ell = M^{\nu} \to \infty$. This equidistribution result holds in fact under explicit diophantine conditions on $\alpha$ (e.g., for $\alpha=\sqrt 2$) provided that $\nu<1$, or $\nu<2$ with additional assumptions on the Fourier coefficients of certain automorphic forms. Finally, we show that for $\nu=2$, our equidistribution theorem implies a recent result of Rudnick and Sarnak on the uniformity of the pair correlation density of the sequence $n^2 \alpha$ modulo one.; Comment: 39 pages

## Equidistribution of Horocyclic Flows on Complete Hyperbolic Surfaces of Finite Area

Hubbard, John H.; Miller, Robyn L.
Tipo: Artigo de Revista Científica
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We provide a self-contained, accessible introduction to Ratner's Equidistribution Theorem in the special case of horocyclic flow on a complete hyperbolic surface of finite area. This equidistribution result was first obtained in the early 1980s by Dani and Smillie and later reappeared as an illustrative special case of Ratner's work on the equidistribution of unipotent flows in homogeneous spaces. We also prove an interesting probabilistic result due to Breuillard: on the modular surface an arbitrary uncentered random walk on the horocycle through almost any point will fail to equidistribute, even though the horocycles are themselves equidistributed. In many aspects of this exposition we are indebted to Bekka and Mayer's more ambitious survey, "Ergodic Theory and Topological Dynamics for Group Actions on Homogeneous Spaces."; Comment: 29 pages, 10 figures

## Equidistribution for higher-rank Abelian actions on Heisenberg nilmanifolds

Flaminio, Livio; Cosentino, Salvatore
Fonte: American Institute of Mathematical Sciences Publicador: American Institute of Mathematical Sciences
Tipo: Artigo de Revista Científica