An Erdős--Fuchs Theorem for Ordered Representation Functions

Abstract

Let $k\ge 2$ be a positive integer. We study concentration results for the ordered representation functions $r_k^{\le }(A,n)=\#\{(a_1\le \cdots \le a_k)\in A^k:a_1+\cdots +a_k=n\}$ and $r_k^{<}(A,n)=\#\{(a_1<\cdots <a_k)\in A^k:a_1+\cdots +a_k=n\}$ for any infinite set of non-negative integers $A$. Our main theorem is an Erdős--Fuchs-type result for both functions: for any $c>0$ and $\star \in \{\le ,<\}$ we show that $$\sum _{j=0}^n(r_k^{\star }(A,j)-c)=o\left(n^{1/4}{\log }^{-1/2}n\right)$$ is not possible. We also show that the mean squared error $$E_{k,c}^{\star }(A,n)=\frac{1}{n}\sum _{j=0}^n(r_k^{\star }(A,j)-c{)}^2$$ satisfies ${\mathrm{limsup}}_{n\to \infty }{E}_{k,c}^{\star }(A,n)>0$. These results extend two theorems for the non-ordered representation function proved by Erdős and Fuchs in the case of $k=2$ (J. of the London Math. Society 1956).

15 pages