Related papers: Improved Bounds for 3-Progressions

The main result in this paper concerns a new five-variable expander. It is proven that for any finite set of real numbers $A$, $$|\{(a_1+a_2+a_3+a_4)^2+\log a_5 :a_1,a_2,a_3,a_4,a_5 \in A \}| \gg \frac{|A|^2}{\log |A|}.$$ This bound is…

In this paper, we prove that the bound \[ \max \{ |8A-7A|,|5f(A)-4f(A)| \} \gg |A|^{\frac{3}{2} + \frac{1}{54}-o(1)} \] holds for all $A \subset \mathbb R$, and for all convex functions $f$ which satisfy an additional technical condition.…

In this note, we show that the method of Croot, Lev, and Pach can be used to bound the size of a subset of $F_q^n$ with no three terms in arithmetic progression by $c^n$ with $c < q$. For $q=3$, the problem of finding the largest subset…

Motivated by applications to matrix multiplication algorithms, Pratt asked (ITCS'24) how large a subset of $[n] \times [n]$ could be without containing a skew-corner: three points $(x,y), (x,y+h),(x+h,y')$ with $h \ne 0$. We prove any skew…

We prove that if $B$ is a set of $N$ positive integers such that $B\cdot B$ contains an arithmetic progression of length $M$, then for some absolute $C > 0$, $$ \pi(M) + C \frac {M^{2/3}}{\log^2 M} \leq N, $$ where $\pi$ is the prime…

A famous conjecture of Graham asserts that every set $A \subseteq \mathbb{Z}_p \setminus \{0\}$ can be ordered so that all partial sums are distinct. Although this conjecture was recently proved for sufficiently large primes by Pham and…

We show that if the difference of two elements of a set $A \subseteq [N]$ is never one less than a prime number, then $|A| = O (N \exp (-c (\log N)^{1/3}))$ for some absolute constant $c>0$.

We give improved lower bounds for binary $3$-query locally correctable codes (3-LCCs) $C \colon \{0,1\}^k \rightarrow \{0,1\}^n$. Specifically, we prove: (1) If $C$ is a linear design 3-LCC, then $n \geq 2^{(1 - o(1))\sqrt{k} }$. A design…

Let $r_k(n)$ denote the maximum cardinality of a set $A \subset \{1,2, \dots, n \}$ such that $A$ does not contain a $k$-term arithmetic progression. In this paper, we give a method of constructing such a set and prove the lower bound…

Suppose that A is a subset of {1,...,N} such that the difference between any two elements of A is never one less than a prime. We show that |A| = O(N exp(-c(log N)^{1/4})) for some absolute c>0.

In the present paper we prove a certain lemma about the structure of "lower level-sets of convolutions", which are sets of the form $\{x \in \Z_N : 1_A*1_A(x) \leq \gamma N\}$ or of the form $\{x \in \Z_N : 1_A*1_A(x) < \gamma N\}$, where…

Let $F$ be a finite field of odd characteristic. We prove that any set $A\subset F$ with $|A|\geq C|F|^{5/6}$ contains a nontrivial quadratic progression $(x, x+y, x+y^2), y\neq 0.$ For prime fields, this improves the previous best-known…

The $3k-4$ Theorem is a classical result which asserts that if $A,\,B\subseteq \mathbb Z$ are finite, nonempty subsets with \begin{equation}\label{hyp}|A+B|=|A|+|B|+r\leq |A|+|B|+\min\{|A|,\,|B|\}-3-\delta,\end{equation} where $\delta=1$ if…

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…

We prove, in particular, that if a subset A of {1, 2,..., N} has no nontrivial solution to the equation x_1+x_2+x_3+x_4+x_5=5y then the cardinality of A is at most N e^{-c(log N)^{1/7-eps}}, where eps>0 is an arbitrary number, and c>0 is an…

The problem of looking for subsets of the natural numbers which contain no 3-term arithmetic progressions has a rich history. Roth's theorem famously shows that any such subset cannot have positive upper density. In contrast, Rankin in 1960…

Let A \subseteq [1,..,N]^2 be a set of cardinality at least N^2/(log log N)^c, where c>0 is an absolute constant. We prove that A contains a triple {(k,m), (k+d,m), (k,m+d)}, where d>0. This theorem is a two-dimensional generalization of…

We prove that for some universal $c$, a non-collinear set of $N>\frac{1}{c}$ points in the Euclidean plane determines at least $c \frac{N}{\log N}$ distinct areas of triangles with one vertex at the origin, as well as at least $c…

This article proves the bound $|\zeta(\frac{1}{2} + it)|\leq 0.732 t^{\frac{1}{6}} \log t$ for $t \geq 2$, which improves on a result by Cheng and Graham. We also show that $|\zeta(\frac{1}{2}+it)|\leq 0.732 |3.3081+it|^{\frac{1}{6}} \log…

We investigate exponential sums over those numbers $\leq x$ all of whose prime factors are $\leq y$. We prove fairly good minor arc estimates, valid whenever $\log^{3}x \leq y \leq x^{1/3}$. Then we prove sharp upper bounds for the $p$-th…