Let be a nontrivial multiplicative character of . We obtain the following results.
(1) Let be given. If is a box satisfying then for we have, denoting a nontrivial multiplicative character, unless is even, is principal on a subfield of size , and .
(2) Assume that so that Then
(3) Let be an interval with , and let be a -spaced set with . Assume that . Then for a nonprincipal multiplicative character , We also slightly improve a result of Karacuba [K3]
Mei-Chu Chang. "On a question of Davenport and Lewis and new character sum bounds in finite fields." Duke Math. J. 145 (3) 409 - 442, 1 December 2008. https://doi.org/10.1215/00127094-2008-056