Additive Representation in Thin Sequences, VII: Restricted Moments of the Number of Representations

Abstract

Let $R_{s,k}(n)$ denote the number of representations of $n$ as the sum of $s$ positive integral $k$th powers. We develop methods to establish asymptotic formulae for moments of $R_{s,k}(n)$ in which $n$ is restricted to a thin sequence consisting of values of a given polynomial. As a first example, we discuss the case of cubes, and present conclusions for sums of 6 and 7 cubes in which the polynomial is quadratic, and for sums of 7 cubes in which the polynomial is cubic. We also consider the case wherein the exponent $k$ is large, and briefly describe corresponding results for the binary Goldbach problem.