On character sums and exponential sums over generalized arithmetic progressions

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Abstract

Let χ (mod q) be a primitive Dirichlet character. In this paper, we prove a uniform upper bound of the character sum ∑a∈A χ(a) over all proper generalized arithmetic progressions A ⊂ ℤ/qℤ of rank r: ∑a∈A χ (n) <<r q1/2(log q)r. This generalizes the classical result by Pólya and Vinogradov. Our method also applies to give a uniform upper bound for the polynomial exponential sum ∑ n∈Aeq(h(n)) (q prime), where h(x) ∈ ℤ[x] is a polynomial of degree 2 ≤ d < q.

Original languageEnglish
Pages (from-to)541-550
Number of pages10
JournalBulletin of the London Mathematical Society
Volume45
Issue number3
DOIs
StatePublished - Jun 2013

ASJC Scopus subject areas

  • Mathematics (all)

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