Characters sums with additive convolutions

Let $\chi(x)$ be a nontrivial multiplicative character over prime modulo $p$, and $A$, $B$ be arbitrary subsets of $\mathbb{Z}/p\mathbb{Z}$ such that $|A+A| \le K|A|$, where $K \ge 1$ be a constant and $|A|,|B|> p^{\,4/9+\varepsilon}$, $\varepsilon>0$.

M.-C. Chang obtained a nontrivial upper bound for the sum
$$ \biggl|\sum_{a\in A,\, b\in B} \chi(a+b)\biggr|\,\ll_{K,\varepsilon}\,|A||B|\cdot p^{-\tau(K,\varepsilon)}, \qquad (1) $$
where $\tau(K,\varepsilon)>0$.

Recently, B. Hanson considered an analog of sum (1) for three sets $A$, $B$ $C$ having no restrictions on its sumsets. Namely, he proved that if $|A|,|B|,|C| > \delta \sqrt{p}$, where $\delta>0$, then
$$ \biggl|\sum_{a\in A,\, b\in B,\, c\in C} \chi(a+b+c)\biggr|\,=\, o_{\delta}\bigl(|A||B||C|\bigr). \qquad (2) $$


Using the almost periodicity lemma of Croot–Sisask as well as new results on sum-products, we refine both (1) and (2).