Related papers: New bounds for dimensions of a set uniformly avoid…

We construct subsets of {1,...,N} of cardinality at least N exp(-C(log N)^{1/(k+1)}) which do not contain arithmetic progressions of length 2^k+1. This extends a result of Behrend (1946) concerning sets which do not contain aritmetic…

Suppose that $d \geq 2$, and that $A \subset [0,1]$ has sufficiently large dimension, $1 - \epsilon_d < \dim_H(A) < 1$. Then for any polynomial $P$ of degree $d$ with no constant term, there exists a point configuration $\{ x, x-t,x-P(t) \}…

We study algorithmic problems on subsets of Euclidean space of low fractal dimension. These spaces are the subject of intensive study in various branches of mathematics, including geometry, topology, and measure theory. There are several…

We establish a new version of Siegel's lemma over a number field $k$, providing a bound on the maximum of heights of basis vectors of a subspace of $k^N$, $N \geq 2$. In addition to the small-height property, the basis vectors we obtain…

Let $f_{s,k}(n)$ be the maximum possible number of $s$-term arithmetic progressions in a sequence $a_1<a_2<\ldots<a_n$ of $n$ integers which contains no $k$-term arithmetic progression. For all integers $k > s \geq 3$, we prove that…

Let $r_5(N)$ be the largest cardinality of a set in $\{1,\ldots,N\}$ which does not contain $5$ elements in arithmetic progression. Then there exists a constant $c\in (0,1)$ such that \[r_5(N)\ll \frac{N}{\exp((\log\log N)^{c})}.\] Our work…

Define $r_4(N)$ to be the largest cardinality of a set $A \subset \{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that \[ r_4(N) \ll N(\log \log N)^{-c}\] for some absolute constant…

We provide upper bounds for the size of subsets of finite fields lacking the polynomial progression $$ x, x+y, ..., x+(m-1)y, x+y^m, ..., x+y^{m+k-1}.$$ These are the first known upper bounds in the polynomial Szemer\'{e}di theorem for the…

Let $d \geq 4$ be a natural number and let $A$ be a finite, non-empty subset of $\mathbb{R}^d$ such that $A$ is not contained in a translate of a hyperplane. In this setting, we show that \[ |A-A| \geq \bigg(2d - 2 + \frac{1}{d-1} \bigg)…

An arc is a subset of $\mathbb F_q^2$ which does not contain any collinear triples. Let $A(q,k)$ denote the number of arcs in $\mathbb F_q^2$ with cardinality $k$. This paper is primarily concerned with estimating the size of $A(q,k)$ when…

We build a bridge from density combinatorics to dimension theory of continued fractions. We establish a fractal transference principle that transfers common properties of subsets of $\mathbb N$ with positive upper density to properties of…

We introduce a new family of fractal dimensions by restricting the set of diameters in the coverings in the usual definition of the Hausdorff dimension. Among others, we prove that this family contains continuum many distinct dimensions,…

We provide quantitative estimates for the supremum of the Hausdorff dimension of sets in the real line which avoid $\varepsilon$-approximations of arithmetic progressions. Some of these estimates are in terms of Szemer\'{e}di bounds. In…

Let p > 4 be a prime. We show that the largest subset of F_p^n with no 4-term arithmetic progressions has cardinality << N(log N)^{-c}, where c = 2^{-22} and N := p^n. A result of this type was claimed in a previous paper by the authors and…

A Sidon set is a set of integers containing no nontrivial solutions to the equation $a+b=c+d$. We improve on the lower bound on the diameter of a Sidon set with $k$ elements: if $k$ is sufficiently large and ${\cal A}$ is a Sidon set with…

We study the fractal dimension of a given subset X of R^{n} such that a motion is possible without crossing X.

We demonstrate $k+1$-term arithmetic progressions in certain subsets of the real line whose "higher-order Fourier dimension" is sufficiently close to 1. This Fourier dimension, introduced in previous work, is a higher-order (in the sense of…

A geometric progression of length $k$ and integer ratio is a set of numbers of the form $\{a,ar,\dots,ar^{k-1}\}$ for some positive real number $a$ and integer $r\geq 2$. For each integer $k \geq 3$, a greedy algorithm is used to construct…

Define $r_4(N)$ to be the largest cardinality of a set $A$ in $\{1,\dots,N\}$ which does not contain four elements in arithmetic progression. In 1998 Gowers proved that $r_4(N) \ll N(\log \log N)^{-c}$ for some absolute constant $c> 0$. In…

We construct Salem sets in $\mathbb{R}/\mathbb{Z}$ of any dimension (including $1$) which do not contain any arithmetic progressions of length $3$. Moreover, the sets can be taken to be Ahlfors regular if the dimension is less than $1$, and…