On character sums and exponential sums over generalized arithmetic progressions. (11th January 2013)

• Record Type: Journal Article
• Authors: Shao, Xuancheng
• Is Part Of: Bulletin of the London Mathematical Society. Volume 45:Part 3(2013:Jun.)
• Page Start: 541
• Page End: 550
• Publication Date: 2013-01-11
• Subjects: Mathematics -- Periodicals; 510
• DOI: 10.1112/blms/bds115
• Languages: English
• ISSNs: 0024-6093

Abstract

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 .