Related papers: Strong Bounds for 3-Progressions

We show that, for any $0<\gamma<1/2$, any $(\alpha,\beta)\in\mathbb{R}^2$ except on a set with Hausdorff dimension about $\sqrt{\gamma}$, any small $0<\varepsilon<1$ and any large $N\in\mathbb{N}$, the number of integers $n\in[1,N]$ such…

Let $\mathbf{P}$ denote the set of prime numbers and, for an appropriate function $h$, define a set $\mathbf{P}_{h}=\{p\in\mathbf{P}: \exists_{n\in\mathbb{N}}\ p=\lfloor h(n)\rfloor\}$. The aim of this paper is to show that every subset of…

For a finite set $A\subset \mathbb{R}^d$, let $\Delta(A)$ denote the spread of $A$, which is the ratio of the maximum pairwise distance to the minimum pairwise distance. For a positive integer $n$, let $\gamma_d(n)$ denote the largest…

We show that the hereditary discrepancy of homogeneous arithmetic progressions is lower bounded by $n^{1/O(\log \log n)}$. This bound is tight up to the constant in the exponent. Our lower bound goes via proving an exponential lower bound…

In 1946, Behrend gave a construction of dense finite sets of integers that do not contain 3-term arithmetic progressions. In 1961, Rankin generalized Behrend's construction to sets avoiding k-term arithmetic progressions, and in 2008 Elkin…

Let $s(n)$ be the number of different remainders $n \bmod k$, where $1 \leq k \leq \lfloor n/2 \rfloor$. This rather natural sequence is sequence A283190 in the OEIS and while some basic facts are known, it seems that surprisingly it has…

Given a set of $n$ real numbers, the 3SUM problem is to decide whether there are three of them that sum to zero. Until a recent breakthrough by Gr{\o}nlund and Pettie [FOCS'14], a simple $\Theta(n^2)$-time deterministic algorithm for this…

We examine the behavior of the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/n\mathbb{Z}$. We prove that if a set is chosen by including each element of $\mathbb{Z}/n\mathbb{Z}$ independently with constant…

The finiteness property is an important arithmetical property of beta-expansions. We exhibit classes of Pisot numbers $\beta$ having the negative finiteness property, that is the set of finite $(-\beta)$-expansions is equal to…

An improved upper bound is obtained for the density of sequences of positive integers that contain no k-term geometric progression.

Suppose that we are given a string $s$ of length $n$ over an alphabet $\{0,1,\ldots,n^{O(1)}\}$ and $\delta$ is the string complexity of $s$, a known compression measure. We describe an index on $s$ with $O(\delta\log\frac{n}{\delta})$…

In 1975 Szemer\'edi proved that a set of integers of positive upper density contains arbitrarily long arithmetic progressions. Bergelson and Leibman showed in 1996 that the common difference of the arithmetic progression can be a square, a…

We establish sharp bounds for the Hausdorff dimension of sets of irrational numbers in $(0,1)$ whose digits in the $N$-expansion are either uniformly bounded or tend to infinity. For sets with digits bounded by an integer $M \ge N$, we…

In a previous paper of the authors, we showed that for any polynomials $P_1,\dots,P_k \in \Z[\mathbf{m}]$ with $P_1(0)=\dots=P_k(0)$ and any subset $A$ of the primes in $[N] = \{1,\dots,N\}$ of relative density at least $\delta>0$, one can…

We prove that for any partition of a set which contains an infinite arithmetic (respectively geometric) progression into two disjoint subsets, at least one of these subsets contains an infinite number of triplets such that each triplet is…

We prove that there are arbitrarily long arithmetic progressions of primes. There are three major ingredients. The first is Szemeredi's theorem, which asserts that any subset of the integers of positive density contains progressions of…

The $3k-4$ Theorem asserts that, if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with $|A|\geq |B|$ and $|A+B|=|A|+|B|+r< |A|+2|B|-3$, then there are arithmetic progressions $P_A$ and $P_B$ of common difference with $X\subseteq…

Detecting commuting patterns or migration patterns in movement data is an important problem in computational movement analysis. Given a trajectory, or set of trajectories, this corresponds to clustering similar subtrajectories. We study…

We investigate additive properties of sets $A,$ where $A=\{a_1,a_2,\ldots ,a_k\}$ is a monotone increasing set of real numbers, and the differences of consecutive elements are all distinct. It is known that $|A+B|\geq c|A||B|^{1/2}$ for any…

Let $G$ be a multiplicative subgroup of the prime field $\mathbb F_p$ of size $|G|> p^{1-\kappa}$ and $r$ an arbitrarily fixed positive integer. Assuming $\kappa=\kappa(r)>0$ and $p$ large enough, it is shown that any proportional subset…