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 \(\max(|A+A|,|A^3+A^3|)\ge|A|^{16/15},\) \(\max(|A+A|,|A^4+A^4|)\ge|A|^{25/24}\) , and \(\max(|A+A|,|A^5+A^5|)\ge|A|^{25/24}\) .