An elementary method for estimating error terms in additive number theory

Abstract

Let R k ( n ) {R_k}(n) denote the number of ways of representing the integers not exceeding n n as the sum of k k members of a given sequence of nonnegative integers. Using only elementary methods, we prove a general theorem from which we deduce that, for every ϵ > 0 \epsilon > 0 , \[ R k ( n ) − c n β ≠ o ( n β ( 1 − β ) ( 1 − 1 / k ) / ( 1 − β + β / k ) − ϵ ) {R_k}(n) - c{n^\beta } \ne o({n^{\beta (1 - \beta )(1 - 1/k)/(1 - \beta + \beta /k) - \epsilon }}) \] where c c is a positive constant and 0 > β > 1 0 > \beta > 1 .