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## Equidistribution and variation of height functions

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.

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## Small scale equidistribution of eigenfunctions on the torus

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 05/08/2015

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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.

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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.

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## Equidistribution estimates for Fekete points on complex manifolds

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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)

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 17/03/2011

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 26/09/2010

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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

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## Equidistribution of points via energy

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 22/07/2013

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Quantitative equidistribution properties of toral eigenfunctions

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 10/03/2015

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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)

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Effective equidistribution of twisted horocycle flows and horocycle maps

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 18/07/2015

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Dynamical Systems#Mathematics - Complex Variables#Mathematics - Number Theory#Primary 37P50, Secondary 11S82

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)

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Sparse equidistribution problems, period bounds, and subconvexity

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Counting and equidistribution in Heisenberg groups

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

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#Mathematics - Differential Geometry#Mathematics - Number Theory#11E39, 11F06, 11N45, 20G20, 53C17, 53C22, 53C55

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

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## Equidistribution over function fields

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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## Equidistribution of Kronecker sequences along closed horocycles

Fonte: Universidade Cornell
Publicador: Universidade Cornell

Tipo: Artigo de Revista Científica

Publicado em 12/11/2002

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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

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

Fonte: Universidade Cornell
Publicador: Universidade Cornell

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

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

Fonte: American Institute of Mathematical Sciences
Publicador: American Institute of Mathematical Sciences

Tipo: Artigo de Revista Científica

Publicado em /11/2015
ENG

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2010 Mathematics Subject Classification: Primary: 37C85, 37A17, 37A45; Secondary: 11K36, 11L07.; We prove quantitative equidistribution results for actions of Abelian subgroups of the (2g + 1)-dimensional Heisenberg group acting on compact (2g + 1)-dimensional homogeneous nilmanifolds. The results are based on the study of the C∞-cohomology of the action of such groups, on tame estimates of the associated cohomological equations and on a renormalization method initially applied by Forni to surface flows and by Forni and the second author to other parabolic flows. As an application we obtain bounds for finite Theta sums defined by real quadratic forms in g variables, generalizing the classical results of Hardy and Littlewood [25, 26] and the optimal result of Fiedler, Jurkat, and Körner [17] to higher dimension.

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