On an application of higher energies to Sidon sets

We show that for any finite set $A$ and an arbitrary $\varepsilon>0$ there is $k=k(\varepsilon)$ such that the higher energy $\mathrm{E}_{k}(A)$ 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 $|A|^{1 / 2+c}$.
Conference ID: 942 0186 5629 Password is a six-digit number, the first three digits of which form the number p + 44, and the last three digits are the number q + 63, where p, q is the largest pair of twin primes less than 1000