On additive shifts of multiplicative almost-subgroups in finite fields

Abstract

We prove that for sets $A,B,C\subset {\mathbb{F}}_p$ with $|A|=|B|=|C|\le \sqrt{p}$ and a fixed $0\ne d\in {\mathbb{F}}_p$ holds $$\max (|AB|,|(A+d)C|)\gg |A{|}^{1+1/26}.$$ In particular, $$|A(A+1)|\gg |A{|}^{1+1/26}$$ and $$\max (|AA|,|(A+1)(A+1)|)\gg |A{|}^{1+1/26}.$$ The first estimate improves the bound by Roche-Newton and Jones. In the general case of a field of order $q=p^m$ we obtain similar estimates with the exponent $1+1/559+o(1)$ under the condition that $AB$ does not have large intersection with any subfield coset, answering a question of Shparlinski. Finally, we prove the estimate $$\left|\sum _{x\in {\mathbb{F}}_q}\psi (x^n)\right|\ll q^{\frac{7-2{\delta }_2}{8}}{n}^{\frac{2+2{\delta }_2}{8}}$$ for Gauss sums over ${\mathbb{F}}_q$, where $\psi$ is a non-trivial additive character and ${\delta }_2=1/56+o(1)$. The estimate gives an improvement over the classical Weil bound when $q^{1/2}\ll n=o\left(q^{29/57+o(1)}\right)$.

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