Weyl’s Sum

Wang Yuan
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Wang Yuan: Academia Sinica, Institute of Mathematics

Chapter Chapter 3 in Diophantine Equations and Inequalities in Algebraic Number Fields, 1991, pp 23-43 from Springer

Abstract: Abstract Let $$ f\left( \lambda \right) = {{\alpha }_{k}}{{\lambda }^{k}} + ... + {{\alpha }_{1}}\lambda $$ be a polynomial of k-th degree with coefficients in J. The sum $$ S(f,T) = S(f(\lambda ),\xi ,T) = \sum\limits_{\lambda \in P(T)} {E(f(\lambda )\xi )} $$ is called a Weyl’s sum. Note that the range of the sum can be replaced by any finite set of integers. Before we state Weyl’s inequality for S(f,T) we begin with Siegel’s generalisation of Dirichlet’s theorem on rational approximations to real numbers.

Keywords: Asymptotic Formula; Diophantine Equation; Integral Ideal; Real Coefficient; Nonzero Integer (search for similar items in EconPapers)
Date: 1991
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DOI: 10.1007/978-3-642-58171-7_3

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