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 $r^{{ \le }}_k({\mathcal {A}},n) = \# \big \{ (a_1 \le \dots \le a_k) \in {\mathcal {A}}^k : a_1+\dots +a_k = n \big \}$ and $r^{{<}}_k({\mathcal {A}},n) = \# \big \{ (a_1< \dots < a_k) \in {\mathcal {A}}^k : a_1+\dots +a_k = n \big \}$ for any infinite set of non-negative integers ${\mathcal {A}}$. Our main theorem is an Erdős–Fuchs-type result for both functions: for any $c > 0$ and $\star \in \{\le ,<\}$ we show that

$$\begin{aligned} \sum _{j = 0}^{n} \Big ( r^{\star }_k ({\mathcal {A}},j) - c \Big )= o\big (n^{1/4}\log ^{-1/2}n\big ) \end{aligned}$$

is not possible. We also show that the mean squared error

$$\begin{aligned} E^\star _{k,c}({\mathcal {A}},n)=\frac{1}{n} \sum _{j = 0}^{n} \Big ( r^{\star }_k({\mathcal {A}},j) - c \Big )^2 \end{aligned}$$

satisfies $\limsup _{n \rightarrow \infty } E^\star _{k,c}({\mathcal {A}},n)>0$. 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).