2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/217395We show that with high probability a random set of size $Θ(n^{1-1/k})$ 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, IntegersCombinatoricstext