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).

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Acknowledgements

The authors thank the anonymous referee for the detailed reading of the manuscript.

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Correspondence to Juanjo Rué.

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G. Cao-Labora: This work was done under the research collaboration Grant 2018 COLAB 00175 from AGAUR (Catalunya).

Juanjo Rué: Supported by the Spanish Ministerio de Economía y Competitividad Project MTM2017-82166-P, and the María de Maetzu research Grant MDM-2014-0445.

Christoph Spiegel: Supported by the Spanish Ministerio de Economía y Competitividad project MTM2017-82166-P and the María de Maetzu research Grant MDM-2014-0445, and by an FPI Grant under the Project MTM2014-54745-P.

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Cao-Labora, G., Rué, J. & Spiegel, C. An Erdős–Fuchs theorem for ordered representation functions. Ramanujan J (2020). https://doi.org/10.1007/s11139-020-00326-2

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Keywords

  • Additive number theory
  • Representation functions
  • Additive basis
  • Erdős–Fuchs Theorem

Mathematics Subject Classification

  • 11B13
  • 11B34
  • 30B10