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 $n=a+a'$ with $a$, $a'\in A$. Denote by $χ_A(n)$ the characteristic function of $A$. Let $b_n>0$ be a sequence satisfying $\limsup_{n\to \infty}b_n<1$. In this paper, we prove some Erd\H os--Fuchs-type theorems about the error terms appearing in approximation formulæ for $R_A(n)=\sum_{k=0}^nχ_A(k)χ_A(n-k)$ and $\sum_{n=0}^NR_A(n)$ having principal terms $\sum_{k=0}^nb_kb_{n-k}$ and $\sum_{n=0}^N\sum_{k=0}^nb_kb_{n-k}$, respectively.