On additive shifts of multiplicative almost-subgroups in finite fields

We prove that for sets $A, B, C \subset \mathbb{F}_p$ with $|A|=|B|=|C| \leq \sqrt{p}$ and a fixed $0 \neq d \in \mathbb{F}_p$ 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 $q^{1/2} \ll n = o\left( q^{29/57 + o(1)} \right)$.