On an application of higher energies to Sidon sets

TL;DR

The paper investigates the structure of finite sets using higher energies and demonstrates that such sets either contain large Sidon-type subsets or have specific structural properties, linking additive and multiplicative combinatorics.

Contribution

It establishes bounds on higher energies of finite sets and applies these results to show the existence of large Sidon-type subsets within any finite set of reals or prime fields.

Findings

Higher energy ${f E}_k(A)$ is bounded by $|A|^{k+ ext{small}}$ unless $A$ has a special structure.

Any finite subset of reals or prime fields contains a large additive or multiplicative Sidon-type subset.

Sets either contain large Sidon-type subsets or exhibit specific structural properties.

Abstract

We show that for any finite set $A$ and an arbitrary $\varepsilon>0$ there is $k=k(\varepsilon)$ such that the higher energy ${\mathsf{E}}_k(A)$ is at most $|A|^{k+\varepsilon}$ 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 $|A|^{1/2+c}$.