On two-point configurations in random set

Abstract

We show that with high probability a random set of size $\Theta (n^{1-1/k})$ of $\{1,...,n\}$ contains two elements $a$ and $a+d^k$, where $d$ is a positive integer. As a consequence, we prove an analogue of Sárközy-Fürstenberg's theorem for random set.

5 pages, accepted to publish, Integers