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On an Application of Higher Energies to Sidon Sets

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Abstract

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

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Correspondence to I. D. Shkredov.

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This work is supported by the Russian Science Foundation under Grant 19–11–00001, https://rscf.ru/pro ject/19-11-00001.

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Shkredov, I.D. On an Application of Higher Energies to Sidon Sets. Combinatorica 43, 329–345 (2023). https://doi.org/10.1007/s00493-023-00013-y

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