On Additive Representation Functions

TL;DR

This paper investigates the properties of additive representation functions of infinite integer sequences, establishing a relationship between the boundedness of the sequence's positive partial sums and the growth of their average positive parts.

Contribution

It provides a new theoretical link between the boundedness of certain partial sums and the average growth of positive parts of these sums in additive number theory.

Findings

If the L-infinity norm of S_k^+ is small, then the L1 norm of S_k^+/k is large.

The paper establishes a quantitative relationship between boundedness and average growth.

Results contribute to understanding the structure of additive representation functions.

Abstract

Let $\A=\{a_1<a_2<a_3.....<a_n<...\}$ be an infinite sequence of integers and let $R_2(n)=|\{(i,j):\ \ a_i+a_j=n;\ \ a_i,a_j\in \A;\ \ i\le j\}|$. We define $S_k=\s_{l=1}^k(R_2(2l)-R_2(2l+1))$. We prove that, if $L^{\infty}$ norm of $S_k^+(=\max\{S_k,0\})$ is small then $L^1$ norm of $\frac{S_k^+}{k}$ is large.