On two-point configurations in random set

TL;DR

This paper demonstrates that random subsets of integers of a certain size almost surely contain specific two-point configurations, extending classical theorems to probabilistic settings.

Contribution

It establishes a probabilistic analogue of Sárközy-Furstenberg's theorem for random sets, showing the presence of k-th power differences.

Findings

Random sets of size Θ(n^{1-1/k}) contain two elements differing by a k-th power with high probability

Extension of classical difference theorems to random subsets of integers

Provides probabilistic bounds for the occurrence of specific configurations

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 S\'ark\"ozy-F\"urstenberg's theorem for random set.