Additive representation functions and discrete convolutions

TL;DR

This paper investigates the error terms in additive representation functions and discrete convolutions for sets of non-negative integers, establishing Erdős–Fuchs-type theorems under certain conditions.

Contribution

It proves new Erdős–Fuchs-type theorems related to approximation errors in additive representation formulas involving characteristic functions and sequences.

Findings

Error bounds for representation functions established

Principal terms for sums characterized

Conditions on sequences with limsup less than 1

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 $\chi_A(n)$ 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 formul\ae\ for $R_A(n)=\sum_{k=0}^n\chi_A(k)\chi_A(n-k)$ 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.