On an Application of Higher Energies to Sidon Sets

We show that for any finite set A and an arbitrary ε >0 there exists k=k(ε ) such that the higher energy _k(A) is at most |A|^k+ε 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 .