On the dimension of additive sets

We show that a non-empty subset of an abelian group with a small edge boundary must be large; in particular, if $A$ and $S$ are finite, non-empty subsets of an abelian group such that $S$ is independent, and the edge boundary of $A$ with respect to $S$ does not exceed $(1-\gamma)|S||A|$ with a real $\gamma\in(0,1]$, then $|A| \ge 4^{(1-1/d)\gamma |S|}$, where $d$ is the smallest order of an element of $S$. Here the constant $4$ is best possible. As a corollary, we derive an...

Any binary string can be associated with a unary predicate $P$ on $\mathbb{N}$. In this paper we investigate subsets named by a predicate $P$ such that the relation $P(x+y)$ has finite VC dimension. This provides a measure of complexity for binary strings with different properties than the standard string complexity function (based on diversity of substrings). We prove that strings of bounded VC dimension are meagre in the topology of the reals, provide simple rules for bound...

Let $0 < \beta \leq \alpha < 1$ and $\kappa > 0$. I prove that there exists $\eta > 0$ such that the following holds for every pair of Borel sets $A,B \subset \mathbb{R}$ with $\dim_{\mathrm{H}} A = \alpha$ and $\dim_{\mathrm{H}} B = \beta$: $$\dim_{\mathrm{H}} \{c \in \mathbb{R} : \dim_{\mathrm{H}} (A + cB) \leq \alpha + \eta\} \leq \tfrac{\alpha - \beta}{1 - \beta} + \kappa.$$ This extends a result of Bourgain from 2010, which contained the case $\alpha = \beta$. The paper ...

We study open point sets in Euclidean spaces $\mathbb{R}^d$ without a pair of points an integral distance apart. By a result of Furstenberg, Katznelson, and Weiss such sets must be of Lebesgue upper density zero. We are interested in how large such sets can be in $d$-dimensional volume. We determine the lower and upper bounds for the volumes of the sets in terms of the number of their connected components and dimension, and also give some exact values. Our problem can be view...

We show that for every large enough integer $N$, there exists an $N$-point subset of $L_1$ such that for every $D>1$, embedding it into $\ell_1^d$ with distortion $D$ requires dimension $d$ at least $N^{\Omega(1/D^2)}$, and that for every $\eps>0$ and large enough integer $N$, there exists an $N$-point subset of $L_1$ such that embedding it into $\ell_1^d$ with distortion $1+\eps$ requires dimension $d$ at least $N^{1-O(1/\log(1/\eps))}$. These results were previously proven ...

We refine the bound on the packing number, originally shown by Haussler, for shallow geometric set systems. Specifically, let $\V$ be a finite set system defined over an $n$-point set $X$; we view $\V$ as a set of indicator vectors over the $n$-dimensional unit cube. A $\delta$-separated set of $\V$ is a subcollection $\W$, s.t. the Hamming distance between each pair $\uu, \vv \in \W$ is greater than $\delta$, where $\delta > 0$ is an integer parameter. The $\delta$-packing n...

Let $A$ and $B$ be Borel subsets of the Euclidean $n$-space with $\dim A + \dim B > n$. This is a survey on the question: what can we say about the Hausdorff dimension of the intersections $A\cap (g(B)+z)$ for generic orthogonal transformations $g$ and translations by $z$.

The distance set $\Delta(E)$ of a set $E$ consists of all non-negative numbers that represent distances between pairs of points in $E$. This paper studies sparse (less than full-dimensional) Borel sets in $\mathbb R^d$, $d \geq 2$ with a focus on properties of their distance sets. Our results are of four types. First, we generalize a classical result of Steinhaus (1920) to Borel sets $E \subseteq [0,1]^d$ with $s$-dimensional Hausdorff content larger than $(1 - \rho)$, for sm...

For $ E\subset \mathbb{F}_q^d$, let $\Delta(E)$ denote the distance set determined by pairs of points in $E$. By using additive energies of sets on a paraboloid, Koh, Pham, Shen, and Vinh (2020) proved that if $E,F\subset \mathbb{F}_q^d $ are subsets with $|E||F|\gg q^{d+\frac{1}{3}}$ then $|\Delta(E)+\Delta(F)|> q/2$. They also proved that the threshold $q^{d+\frac{1}{3}}$ is sharp when $|E|=|F|$. In this paper, we provide an improvement of this result in the unbalanced case...

Let A be an asymptotic basis for N and X a finite subset of A such that A\X is still an asymptotic basis. Farhi recently proved a new batch of upper bounds for the order of A\X in terms of the order of A and a variety of parameters related to the set X. He posed two questions concerning possible improvements to his bounds. In this note, we answer both questions.