On an application of higher energies to Sidon sets

Abstract

We show that for any finite set $A$ and an arbitrary $\epsilon >0$ there is $k=k(\epsilon )$ such that the higher energy ${\mathsf{E}}_k(A)$ is at most $|A{|}^{k+\epsilon }$ unless $A$ has a very specific structure. As an application we obtain that any finite subset $A$ of the real numbers or the prime field either contains an additive Sidon--type subset of size $|A{|}^{1/2+c}$ or a multiplicative Sidon--type subset of size $|A{|}^{1/2+c}$.

arXiv comment15 pages