On a question of Davenport and Lewis and new character sum bounds in finite fields

Let $\chi$ be a nontrivial multiplicative character of $\Bbb F_{p^n}$ . We obtain the following results. ¶ (1) Let $\varepsilon>0$ be given. If $B=\big\{ \sum _{j=1}^n x_j \omega_j \;: x_j \in [N_j+1, N_j+H_j]\cap \Bbb Z , j=1,\ldots, n %\text{ for all } j \big\}$ is a box satisfying ${\mathop\Pi\limits}_{j=1}^{n}H_j>p^{({2}/{5}+\varepsilon)n},$ then for $p>p(\varepsilon)$ we have, denoting $\chi$ a nontrivial multiplicative character, \[ \Big| \sum_{x \in B} \chi(x) \Big| \ll_np^{-{\varepsilon^{2}}/4} |B| \] unless $n$ is even, $\chi$ is principal on a subfield $F_2$ of size $p^{n/2}$ , and $\max_\xi\!\!|B\cap \xi F_2| >p^{-\varepsilon}|B|$ . ¶ (2) Assume that $A, B \subset \Bbb F_p$ so that \[|A|> p^{(4/9)+\varepsilon},\qquad |B|> p^{(4/9)+\varepsilon},\qquad |B+B| \lt K|B|.\] Then \[\Big|\sum_{x\in A, y\in B} \chi(x+y)\Big| \lt p^{-\tau}|A|\;|B|.\] ¶ (3) Let $I\subset \Bbb F_p$ be an interval with $|I|=p^{\beta}$ , and let $\mathcal D\subset \Bbb F_p$ be a $p^\beta$ -spaced set with $|\mathcal D|=p^\sigma$ . Assume that $2\beta+\sigma-{\beta\sigma}/{(1-\beta)}> 1/2+\delta$ . Then for a nonprincipal multiplicative character $\chi$ , \[\Big|\sum_{x\in I, y\in \mathcal D}\chi(x+y)\Big| \lt p^{-{\delta^2}/{12}}|I|\;\;|\mathcal D|.\] We also slightly improve a result of Karacuba [K3]