Related papers: A Marstrand theorem for subsets of integers

Let $E, F\subset \R^d$ be two self-similar sets. Under mild conditions, we show that $F$ can be $C^1$-embedded into $E$ if and only if it can be affinely embedded into $E$; furthermore if $F$ can not be affinely embedded into $E$, then the…

In this article, we study the structure of the difference set $E - E$ for subsets $E \subseteq \mathbb{Z}^2$ of positive upper Banach density. Fish asked in [Proc. Amer. Math. Soc. 146 (2018), 3449-3453] whether, for every such set $E$,…

Let $1 \leq m < s \leq n$ and let $A \subseteq \mathbb{R}^n$ be a Borel set of with $s$-dimensional Hausdorff measure $\mathcal{H}^s(A) > 0$. The classical Marstrand slicing theorem states that, for almost every $m$-dimensional subspace $V…

Fix a prime $p\geq 11$. We show that there exists a positive integer $m$ such that any subset of $\mathbb{F}_p^n\times\mathbb{F}_p^n$ containing no nontrivial configurations of the form $(x,y),(x,y+z),(x,y+2z),(x+z,y)$ must have density…

In this paper we show that every set $A \subset \mathbb{N}$ with positive density contains $B+C$ for some pair $B,C$ of infinite subsets of $\mathbb{N}$, settling a conjecture of Erd\H{o}s. The proof features two different decompositions of…

The main result of this paper is the following: if F is any field and R is any F-subalgebra of the algebra of nxn matrices over F with Lie nilpotence index m, then the F-dimension of R is less or equal than M(m+1,n), where M(m+1,n) is the…

The theory of uniform approximation of real numbers motivates the study of products of consecutive partial quotients in regular continued fractions. For any non-decreasing positive function $\varphi:\mathbb{N}\to [2,\infty)$, we determine…

We show that a finite set of integers $A \subseteq \mathbb{Z}$ with $|A+A| \le K |A|$ contains a large piece $X \subseteq A$ with Fre\u{i}man dimension $O(\log K)$, where large means $|A|/|X| \ll \exp(O(\log^2 K))$. This can be thought of…

For a compact set $K\subset \mathbb{R}^1$ and a family $\{C_\lambda\}_{\lambda\in J}$ of dynamically defined Cantor sets sufficiently close to affine with $\text{dim}_H\, K+\text{dim}_H\, C_\lambda>1$ for all $\lambda\in J$, under natural…

Suppose $\Lambda$ is a discrete infinite set of nonnegative real numbers. We say that $ {\Lambda}$ is type $1$ if the series $s(x)=\sum_{\lambda\in\Lambda}f(x+\lambda)$ satisfies a zero-one law. This means that for any non-negative…

We show any subset $A\subset\mathbb{N}$ with positive upper Banach density contains the pattern $\{m,m+[n\alpha],\dots,m+k[n\alpha]\}$, for some $m\in\mathbb{N}$ and $n=p-1$ for some prime $p$, where…

Let $\mathcal{F}$ be a countable collection of functions $f$ defined on the integers with integer values, such that for every $f\in \mathcal{F}$, $f(n)\to +\infty$ as $n\to +\infty$. This paper primarily investigates the Hausdorff dimension…

We generalize a result by Cook, Magyar, and Pramanik [3] on three-term arithmetic progressions in subsets of $\mathbb{R}^d$ to corners in subsets of $\mathbb{R}^d\times\mathbb{R}^d$. More precisely, if $1<p<\infty$, $p\neq 2$, and $d$ is…

We introduce the notion of index of summability for pairs of Banach spaces; for Banach spaces E; F, this index plays the role of a kind of measure of how the m-homogeneous polynomials from E to F are far from being absolutely summing. In…

While studying set function properties of Lebesgue measure, F. Barthe and M. Madiman proved that Lebesgue measure is fractionally superadditive on compact sets in $\mathbb{R}^n$. In doing this they proved a fractional generalization of the…

For every positive integer $d$, we show that there must exist an absolute constant $c > 0$ such that the following holds: for any integer $n \geq cd^{7}$ and any red-blue coloring of the one-dimensional subspaces of $\mathbb{F}_{2}^{n}$,…

In this paper we use the theory of computing to study fractal dimensions of projections in Euclidean spaces. A fundamental result in fractal geometry is Marstrand's projection theorem, which shows that for every analytic set E, for almost…

Suppose $E \subseteq \mathbb{R}$ is nowhere dense. If $(\mathbb{R},<,+,(x \mapsto \lambda x)_{\lambda \in \mathbb{R} }, E)$ does not define every bounded Borel subset of every $\mathbb{R}^n$ then for every $s > 0$ we have $$ | \{ k \in…

Let $F \subset \mathbb{R}^{2}$, and let $\dim_{\mathrm{A}}$ stand for Assouad dimension. I prove that $\dim_{\mathrm{A}} \pi_{e}(F) \geq \min\{\dim_{\mathrm{A}} F,1\}$ for all $e \in S^{1}$ outside of a set of Hausdorff dimension zero. This…

A beautiful theorem of Zeckendorf states that every integer can be written uniquely as the sum of non-consecutive Fibonacci numbers $\{ F_i \}_{i = 1}^{\infty}$. A set $S \subset \mathbb{Z}$ is said to satisfy Benford's law if the density…