Additive representation functions and discrete convolutions

Abstract

For a set $A$ of non-negative integers, let $R_A(n)$ denote the number of solutions to the equation $n=a+a^{\prime }$ with $a$, $a^{\prime }\in A$. Denote by ${\chi }_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying ${\mathrm{limsup}}_{n\to \infty }{b}_n<1$. In this paper, we prove some Erd\H os--Fuchs-type theorems about the error terms appearing in approximation formulæ for $R_A(n)={\sum }_{k=0}^n{\chi }_A(k){\chi }_A(n-k)$ and ${\sum }_{n=0}^N{R}_A(n)$ having principal terms ${\sum }_{k=0}^n{b}_k{b}_{n-k}$ and ${\sum }_{n=0}^N{\sum }_{k=0}^n{b}_k{b}_{n-k}$, respectively.

13 pages