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

Abstract

Let χ be a nontrivial multiplicative character of Fpn. We obtain the following results. (1). Let ε> 0 be given. If B = {∑nj=1 xjωj: xj ∈ [Nj+1, Nj+Hj]∩Z, j = 1,..., n} is a box satisfying n Π j=1