Related papers: Finite configurations in sparse sets

For given $\epsilon>0$ and $b\in\mathbb{R}^m$, we say that a real $m\times n$ matrix $A$ is $\epsilon$-badly approximable for the target $b$ if $$\liminf_{q\in\mathbb{Z}^n, \|q\|\to\infty} \|q\|^n \langle Aq-b \rangle^m \geq \epsilon,$$…

We extend a result, due to Mattila and Sjolin, which says that if the Hausdorff dimension of a compact set $E \subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then the distance set $\Delta(E)=\{|x-y|: x,y \in E \}$ contains…

Let Q_i, i=1,...,t, be real nondegenerate indefinite quadratic forms in d variables. We investigate under what conditions the closure of the set {(Q_1(x),...,Q_t(x)): x\in Z^d-{0}} contains (0,..,0). As a corollary, we deduce several…

Two subanalytic subsets of R^n are called s-equivalent at a common point P if the Hausdorff distance between their intersections with the sphere centered at P of radius r vanishes of order greater than s when r tends to 0. In this paper we…

A Besicovitch-Rado-Kinney (BRK) set in $\mathbb{R}^n$ is a Borel set that contains a $(n-1)$-dimensional sphere of radius $r$, for each $r>0$. It is known that such sets have Hausdorff dimension $n$ from the work of Kolasa and Wolff. In…

In the present work we establish a Bowen-type formula for the Hausdorff dimension of shrinking-target sets for non-autonomous conformal iterated function systems in arbitrary dimensions and satisfying certain conditions. In the case of…

Using principles of the theory of smoothness spaces we give systematic constructions of scales of inverse-closed subalgebras of a given Banach algebra with the action of a d-parameter automorphism group. In particular we obtain the…

Let $\mathbf{A}$ be an $n\times n$-matrix over $\mathbb{F}_2$ whose every entry equals $1$ with probability $d/n$ independently for a fixed $d>0$. Draw a vector $\mathbf{y}$ randomly from the column space of $\mathbf{A}$. It is a simple…

This paper concerns the expansion of the real ordered additive group by a predicate for a subset of $[0,1]$ whose base-$r$ representations are recognized by a B\"uchi automaton. In the case that this predicate is closed, a dichotomy is…

We prove that for all $b$, the Hausdorff dimension of the set of $m \times n$ matrices $\epsilon$-badly approximable for the target $b$ is not full. The doubly metric case follows. It was known that for almost every matrix $A$, the…

If $E \subset \mathbb{R}^2$ is a compact set of Hausdorff dimension greater than $5/4$, we prove that there is a point $x \in E$ so that the set of distances $\{ |x-y| \}_{y \in E}$ has positive Lebesgue measure.

We develop a method for the rigorous estimation of Hausdorff dimensions of limit sets produced by continued fraction iterated function systems. Our method is based on the approximation of a Perron-Frobenius operator using the finite element…

Let $M$ be a compact $d$-dimensional Riemannian manifold without a boundary. Given $E \subset M$, let $\Delta_{\rho}(E)=\{\rho(x,y): x,y \in E \}$, where $\rho$ is the Riemannian metric on $M$. Let $\Delta_{\rho}^x$ denote the pinned…

An important theorem of geometric measure theory (first proved by Besicovitch and Davies for Euclidean space) says that every analytic set of non-zero $s$-dimensional Hausdorff measure $\mathcal H^s$ contains a closed subset of non-zero…

Consider a finite Blaschke product $f$ with $f(0) = 0$ which is not a rotation and denote by $f^n$ its $n$-th iterate. Given a sequence $\{a_n\}$ of complex numbers, consider the series $F(z) = \sum_n a_n f^n(z).$ We show that for any $w…

Let $B$ be a $d$-dimensional Gaussian process on $\mathbb{R}$, where the component are independents copies of a scalar Gaussian process $B_0$ on $\mathbb{R}_+$ with a given general variance function…

We give a generalization of the geometric estimate used by Hart and the second author in their 2008 work on sums and products in finite fields. Their result concerned level sets of non-degenerate bilinear forms over finite fields, while in…

We consider the problem of minimising an inhomogeneous anisotropic elliptic functional in a class of closed $m$ dimensional subsets of $\mathbf{R}^n$ which is stable under taking smooth deformations homotopic to the identity and under local…

It is a longstanding conjecture that given a subset $E$ of a metric space, if $E$ has finite Hausdorff measure in dimension $\alpha\ge 0$ and $\mathscr{H}^\alpha\llcorner E$ has unit density almost everywhere, then $E$ is an…

For an m by n real matrix A, we investigate the set of badly approximable targets for A as a subset of the m-torus. It is well known that this set is large in the sense that it is dense and has full Hausdorff dimension. We investigate the…