On a Question of Davenport and Lewis and New Character Sum Bounds in Finite Fields

Let χ be a nontrivial multiplicative character of Fpn . We obtain the following results. (1). Let ε > 0 be given. If B = {Pnj=1 xjωj : xj ∈ [Nj + 1, Nj + Hj ] ∩ Z, j = 1, . . . , n} is a box satisfying n Π j=1 Hj > p ( 2 5, then for p > p(ε) we have, denoting χ a nontrivial multiplicative character | X x∈B χ(x)| ¿n p− ε 2 4 |B| unless n is even, χ is principal on a subfield F2 of size pn/2 and maxξ |B∩ξF2| > p−ε|B|. (2). Assume A, B ⊂ Fp such that |A| > p 9+ε, |B| > p 9+ε, |B + B| < K|B|. Then X x∈A,y∈B χ(x + y) < p−τ |A| |B|. (3). Let I ⊂ Fp be an interval with |I| = pβ and let D ⊂ Fp be a pβspaced set with |D| = pσ. Assume β > 1 4 − σ 4(1−σ) +δ. Then for a non-principal multiplicative character χ X x∈I,y∈D χ(x + y) < p− δ2 4 |I| |D|. We also improve a result of Karacuba. Typeset by AMS-TEX