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

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

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).

令\（k \ ge 2 \）为正整数。我们研究有序表示函数\（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 \} \）和\（r ^ {{<<}} _ k（{\ mathcal {A}}，n）= \＃\ big \ {（a_1 <\ dots <a_k）\ in {\ mathcal {A}} ^ k：a_1 + \ dots + a_k = n \ big \} \）对于任意无限数量的非负整数\（{\ mathcal { A}} \）。我们的主要定理是两个函数的Erdős-Fuchs型结果：对于任何\（c> 0 \）和\（\ star \ in \ {\ le，<\} \），我们证明

不可能。我们还表明均方误差

满足\（\ limsup _ {n \ rightarrow \ infty} E ^ \ star _ {k，c}（{\ mathcal {A}}，n）> 0 \）。这些结果扩展了由Erdős和Fuchs在\（k = 2 \）的情况下证明的无序表示函数的两个定理（J. of London Math。Society 1956）。