On character sums and exponential sums over generalized arithmetic progressions. (11th January 2013) - Shao, Xuancheng
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/ 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 ∈ A e q ( h ( n )) ( q prime), where h ( x )∈ℤ[ x ] is a polynomial of degree 2⩽ d < q .