On character sums and exponential sums over generalized arithmetic progressions

Abstract

Let χ (mod q) be a primitive Dirichlet character. In this paper, we prove a uniform upper bound of the character sum ∑a∈Aχ(a) over all proper generalized arithmetic progressions A⊂ℤ/q ℤ of rank r: ∑ n ∊ A χ ( n ) ≪ r q 1 / 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.