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Michael Boshernitzan

Publications and source records attributed to Michael Boshernitzan.

At least 19 recordsLinked to original sources
arXiv2019

Ergodic Schrödinger Operators in the Infinite Measure Setting

We develop the basic theory of ergodic Schrödinger operators, which is well known for ergodic probability measures, in the case of a base dynamics on an infinite measure space. This includes the almost sure constancy of the spectrum and the spectral type, the definition and discussion of the density of states measure and the Lyapunov exponent, as well as a version of the Pastur--Ishii theorem. We also give some counterexamples that demonstrate that some results do not extend from the finite measure case to the infinite measure case. These examples are based on some constructions in infinite ergodic theory that may be of independent interest.

math.SP↗
arXiv2018

On Visibility Problems with an Infinite Discrete, set of Obstacles

This paper studies visibility problems in Euclidean spaces $\mathbb{R}^d$ where the obstacles are the points of infinite discrete sets $Y\subseteq\mathbb{R}^d$. A point $x\in\mathbb{R}^d$ is called $\varepsilon$-visible for $Y$ (notation: $x\in\mathbf{vis}(Y, \varepsilon))$ if there exists a ray $L\subseteq\mathbb{R}^d$ emanating from $x$ such that $||y-z||\geq\varepsilon$, for all $y\in Y\setminus\{x\}$ and $z\in L$. A point $x\in\mathbb{R}^d$ is called visible for $Y$ (notation: $x\in\mathbf{vis}(Y))$ if $x\in\mathbf{vis}(Y, \varepsilon))$, for some $\varepsilon>0$.\\ Our main result is the following. For every $\varepsilon>0$ and every relatively dense set $Y\subseteq\mathbb{R}^2$, $\mathbf{vis}(Y, \varepsilon))\neq\mathbb{R}^2$. This result generalizes a theorem of Dumitrescu and Jiang, which settled Mitchell's dark forest conjecture. On the other hand, we show that there exists a relatively dense subset $Y\subseteq \mathbb{Z}^d$ such that $\mathbf{vis}(Y)=\mathbb{R}^d$. (One easily verifies that $\mathbf{vis}(\mathbb{Z}^d)=\mathbb{R}^d\setminus\mathbb{Z}^d$, for all $d\geq 2$). We derive a number of other results clarifying how the size of a sets $Y\subseteq\mathbb{R}^d$ may affect the sets $\mathbf{vis}(Y)$ and $\mathbf{vis}(Y,\varepsilon)$. We present a Ramsey type result concerning uniformly separated subsets of $\mathbb{R}^2$ whose growth is faster than linear.

math.MG↗
arXiv2017

From a Packing Problem to Quantitative Recurrence in $[0,1]$ and the Lagrange Spectrum of Interval Exchanges

This article provides optimal constants for two quantitative recurrence problems. First of all for recurrence of maps of the interval [0,1] that preserve the Lebesgue measure and on the other hand Lagrange spectrum of interval exchange transformations. Both results are based on a non-conventional packing problem in the plane with respect to the "pseudo-norm" N(x,y) = sqrt(|xy|).

math.DS↗
arXiv2016

Under recurrence in the Khintchine recurrence theorem

The Khintchine recurrence theorem asserts that on a measure preserving system, for every set $A$ and $\varepsilon>0$, we have $μ(A\cap T^{-n}A)\geq μ(A)^2-\varepsilon$ for infinitely many $n\in \mathbb{N}$. We show that there are systems having under-recurrent sets $A$, in the sense that the inequality $μ(A\cap T^{-n}A)< μ(A)^2$ holds for every $n\in \mathbb{N}$. In particular, all ergodic systems of positive entropy have under-recurrent sets. On the other hand, answering a question of V.~Bergelson, we show that not all mixing systems have under-recurrent sets. We also study variants of these problems where the previous strict inequality is reversed, and deduce that under-recurrence is a much more rare phenomenon than over-recurrence. Finally, we study related problems pertaining to multiple recurrence and derive some interesting combinatorial consequences.

math.DS↗
arXiv2013

A dichotomy for the stability of arithmetic progressions

Let H stand for the set of homeomorphisms on [0,1]. We prove the following dichotomy for Borel subsets A of [0,1]: either there exists a homeomorphism f in H such that the image f(A) contains no 3-term arithmetic progressions; or, for every f in H, the image f(A) contains arithmetic progressions of arbitrary finite length. In fact, we show that the first alternative holds if and only if the set A is meager (a countable union of nowhere dense sets).

math.DS↗
arXiv2012

Approximate embedding of large polygons into $Z^2$

Let $Z^2$ denote the standard lattice in the plane $R^2$. We prove that given a finite subset $S\subset R^2$ and $\eps>0$, then for all sufficiently large dilations $t>0$ there exists a rotation $ρ\colon R^2\to R^2$ around the origin such that $\dist(ρ(tz),Z^2)<\eps$, for all $z\in S$. The result, in a larger generality, has been proved in 2006 by Tamar Ziegler (improving earlier results by Furstenberg, Katznelson, Weiss). The proof presented in the paper is short and self-contained.

math.NT↗
arXiv2012

Subgroup of interval exchanges generated by torsion elements and rotations

Denote by $G$ the group of interval exchange transformations (IETs) on the unit interval. Let $G_{per}\subset G$ be the subgroup generated by torsion elements in $G$ (periodic IETs), and let $G_{rot}\subset G$ be the subset of 2-IETs (rotations). The elements of the subgroup $G_1=< G_{per},G_{rot}>\subset G$ (generated by the sets $G_{per}$ and $G_{rot}$) are characterized constructively in terms of their Sah-Arnoux-Fathi (SAF) invariant. The characterization implies that a non-rotation type 3-IET lies in $G_1$ if and only if the lengths of its exchanged intervals are linearly dependent over $\Q$. In particular, $G_1\subsetneq G$. The main tools used in the paper are the SAF invariant and a recent result by Y. Vorobets that $G_{per}$ coincides with the commutator subgroup of $G$.

math.DS↗
arXiv2012

A dichotomy for projections of planar sets

We prove that most one-dimensional projections of a discrete subset of a plane are either dense in R (the real line), or form a discrete subset of R. More precisely, the set E of exceptional directions (for which the indicated dichotomy fails) is a meager subset of the unit circle T of Lebesgue measure 0. The set E however does not need to be small in the sense of Hausdorff dimension.

math.MG↗
arXiv2012

Linearly repetitive Delone sets Delone sets with finite local complexity: Linear repetitivity versus positivity of weights

We consider Delone sets with finite local complexity. We characterize validity of a subadditive ergodic theorem by uniform positivity of certain weights. The latter can be considered to be an averaged version of linear repetitivity. In this context, we show that linear repetitivity is equivalent to positivity of weights combined with a certain balancedness of the shape of return patterns.

math.CO↗
arXiv2011

Borel-Cantelli sequences

A sequence $\{x_{n}\}_1^\infty$ in $[0,1)$ is called Borel-Cantelli (BC) if for all non-increasing sequences of positive real numbers $\{a_n\}$ with $\underset{i=1}{\overset{\infty}{\sum}}a_i=\infty$ the set \[\underset{k=1}{\overset{\infty}{\cap}} \underset{n=k}{\overset{\infty}{\cup}} B(x_n, a_n))=\{x\in[0,1)\mid |x_n-x|<a_n \text{for} \infty \text{many}n\geq1\}\] has full Lebesgue measure. (To put it informally, BC sequences are sequences for which a natural converse to the Borel-Cantelli Theorem holds). The notion of BC sequences is motivated by the Monotone Shrinking Target Property for dynamical systems, but our approach is from a geometric rather than dynamical perspective. A sufficient condition, a necessary condition and a necessary and sufficient condition for a sequence to be BC are established. A number of examples of BC and not BC sequences are presented. The property of a sequence to be BC is a delicate diophantine property. For example, the orbits of a pseudo-Anosoff IET (interval exchange transformation) are BC while the orbits of a "generic" IET are not. The notion of BC sequences is extended to more general spaces.

math.DS↗
arXiv2011

Diophantine properties of IETs and general systems: Quantitative proximality and connectivity

We present shrinking targets results for general systems with the emphasis on applications for IETs (interval exchange transformations) $(J,T)$, $J=[0,1)$. In particular, we prove that if an IET $(J,T)$ is ergodic (relative to the Lebesgue measure $\lam$), then the equality \[ \liminf_{n\to\infty}\limits n |T^n(x)-y|=0 \tag{A1} \] holds for $\lam\ttimes\lam$-a. a. $(x,y)\in J^2$. The ergodicity assumption is essential: the result does not extend to all minimal IETs. The factor $n$ in (A1) is optimal (e. g., it cannot be replaced by $n \ln(\ln(\ln n))$. On the other hand, for Lebesgue almost all 3-IETs $(J,T)$ we prove that for all $\eps>0$ \[ \liminf_{n\to\infty}\limits n^\eps |T^n(x)-T^n(y)|= \infty,\quad \text{for Lebesgue a. a.} (x,y)\in J^2. \tag{A2} \] This should be contrasted with the equality $ \liminf_{n\to\infty}\limits |T^n(x)-T^n(y)|=0, $ for a. a. $(x,y)\in J^2$, which holds since $(J^2, T\times T)$ is ergodic (because generic 3-IETs $(J,T)$ are weakly mixing). We also prove that no 3-IET is strongly topologically mixing.

math.DS↗
arXiv2009

Continued fractions and heavy sequences

We initiate the study of the sets $H(c)$, $0 =x-[x]$ stands for the fractional part of $x\in \mathbb R$. We prove that, for rational $c$, the sets $H(c)$ are of positive Hausdorff dimension and, in particular, are uncountable. For integers $m\geq1$, we obtain a surprising characterization of the numbers $α\in H_m= H(\frac1m)$ in terms of their continued fraction expansions: The odd entries (partial quotients) of these expansions are divisible by $m$. The characterization implies that $x\in H_m$ if and only if $\frac 1{mx} \in H_m$, for $x>0$. We are unaware of a direct proof of this equivalence, without making a use of the mentioned characterization of the sets $H_m$. We also introduce the dual sets $\hat H_m$ of reals $y$ for which the sequence of integers $\big([ky]\big)_{k\geq1}$ consistently hits the set $m\mathbb Z$ with the at least expected frequency $\frac1m$ and establish the connection with the sets $H_m$: {2mm} If $xy=m$ for $x,y>0$, then $x\in H_m$ if and only if $y\in \hat H_m$. The motivation for the present study comes from Y. Peres's ergodic lemma.

math.NT↗
arXiv2009

Almost-Everywhere Convergence and Polynomials

Denote by $Γ$ the set of pointwise good sequences. Those are sequences of real numbers $(a_k)$ such that for any measure preserving flow $(U_t)_{t\in \mathbb R}$ on a probability space and for any $f\in L^\infty$, the averages $\frac{1}{n} \sum_{k=1}^{n} f(U_{a_k}x) $ converge almost everywhere. We prove the following two results. [1.] If $f: (0,\infty)\to\mathbb R$ is continuous and if $\big(f(ku+v)\big)_{k\geq 1}\inΓ$ for all $u, v>0$, then $f$ is a polynomial on some subinterval $J\subset (0,\infty)$ of positive length. [2.] If $f: [0,\infty)\to\mathbb R$ is real analytic and if $\big(f(ku)\big)_{k\geq 1}\inΓ$ for all $u>0$, then $f$ is a polynomial on the whole domain $[0,\infty)$. These results can be viewed as converses of Bourgain's polynomial ergodic theorem which claims that every polynomial sequence lies in $Γ$.

math.DS↗
arXiv2009

On two recurrence problems

We review some aspects of recurrence in topological dynamics and focus on two open problems. The first is an old one concerning the relation between Poincare and Birkhoff recurrence; the second, due to Boshernitzan, is about moving recurrence. We provide a partial answer to a topological version of the moving recurrence problem.

math.DS↗
arXiv2008

Pinned Repetitions in Symbolic Flows: Preliminary Results

We consider symbolic flows over finite alphabets and study certain kinds of repetitions in these sequences. Positive and negative results for the existence of such repetitions are given for codings of interval exchange transformations and codings of quadratic polynomials.

math.DS↗
arXiv2007

The Repetition Property for Sequences on Tori Generated by Polynomials or Skew-Shifts

The repetition property of a sequence in a metric space, a notion introduced by us in an earlier paper, is of importance in the spectral analysis of ergodic Schrödinger operators. It may be used to exclude eigenvalues for such operators. In this paper we study the question of when a sequence on a torus that is generated by a polynomial or a skew-shift has the repetition property. This provides classes of ergodic Schrödinger operators with potentials generated by skew-shifts on tori that have, contrary to earlier belief, no eigenvalues.

math.DS↗