On the Energy of Roots

An estimate of the additive energy of roots modulo a prime for sets with small doubling that has recently been obtained by Zaharescu, Kerr, Shkredov, and Shparlinskii is improved. The problem of determining the maximum cardinalities of the sets $|A+A|$ and $|f(A)+f(A)|$, where $f$ is a polynomial of small degree and $A$ is a subset of a finite field whose size is sufficiently small in comparison with the characteristic of the field, is also considered. In particular, it is proved that
$$ \max(|A+A|,|A^3+A^3|)\geqslant |A|^{16/15}, $$
$\max(|A+A|,|A^4+A^4|)\geqslant |A|^{25/24}$, and $\max(|A+A|,|A^5+A^5|)\geqslant |A|^{25/24}$.