On two-point configurations in random set

We show that with high probability a random set of size $\mathrm{\Theta }(n^{1-1\mathrm{/}k})$ of $\{1,\mathrm{.}\mathrm{.}\mathrm{.},n\}$ 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.