Duke Mathematical Journal

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

Mei-Chu Chang

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Abstract

Let χ be a nontrivial multiplicative character of Fpn. We obtain the following results.

(1) Let ϵ>0 be given. If B={j=1nxjωj :xj[Nj+1,Nj+Hj]Z,j=1,,n} is a box satisfying Πj=1nHj>p(2/5+ϵ)n, then for p>p(ϵ) we have, denoting χ a nontrivial multiplicative character, |xBχ(x)|np-ϵ2/4|B| unless n is even, χ is principal on a subfield F2 of size pn/2, and maxξ|BξF2|>p-ϵ|B|.

(2) Assume that A,BFp so that |A|>p(4/9)+ϵ,    |B|>p(4/9)+ϵ,    |B+B|<K|B|. Then |xA,yBχ(x+y)|<p-τ|A| |B|.

(3) Let IFp be an interval with |I|=pβ, and let DFp be a pβ-spaced set with |D|=pσ. Assume that 2β+σ-βσ/(1-β)>1/2+δ. Then for a nonprincipal multiplicative character χ, |xI,yDχ(x+y)|<p-δ2/12|I|  |D|. We also slightly improve a result of Karacuba [K3]

Article information

Source
Duke Math. J., Volume 145, Number 3 (2008), 409-442.

Dates
First available in Project Euclid: 15 December 2008

Permanent link to this document
https://projecteuclid.org/euclid.dmj/1229349902

Digital Object Identifier
doi:10.1215/00127094-2008-056

Mathematical Reviews number (MathSciNet)
MR2462111

Zentralblatt MATH identifier
1241.11137

Subjects
Primary: 11L40: Estimates on character sums 11L26: Sums over arbitrary intervals
Secondary: 11A07: Congruences; primitive roots; residue systems 11B75: Other combinatorial number theory

Citation

Chang, Mei-Chu. On a question of Davenport and Lewis and new character sum bounds in finite fields. Duke Math. J. 145 (2008), no. 3, 409--442. doi:10.1215/00127094-2008-056. https://projecteuclid.org/euclid.dmj/1229349902


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