Discrete spheres and arithmetic progressions in product sets
Volume 178 / 2017
Abstract
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 $C \gt 0$, $$ \pi(M) + C \frac {M^{2/3}}{\log^2 M} \leq N, $$ where $\pi$ is the prime counting function. This improves on previously known bounds of the form $N = \Omega(\pi(M))$ and gives a bound which is sharp up to the second order term, as Pách and Sándor gave an example for which $$ N \lt \pi(M)+ O\biggl(\frac {M^{2/3}}{\log^2 M} \biggr). $$ The main new tool is a reduction of the original problem to the question of approximate additive decomposition of the $3$-sphere in $\mathbb{F}_3^n$ which is the set of 0-1 vectors with exactly three non-zero coordinates. Namely, we prove that such a set cannot have an additive basis of order two of size less than $c n^2$ with absolute constant $c \gt 0$.
