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On infinite arithmetic progressions in sumsets

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Abstract

Let k be a positive integer. Denote by D1/k the least integer d such that for every set A of nonnegative integers with the lower density l/k, the set (k + l)A contains an infinite arithmetic progression with difference at most d, where (k +l)A is the set of all sums of k + l elements (not necessarily distinct) of A. Chen and Li (2019) conjectured that D1/k = k2 + o(k2). The purpose of this paper is to confirm the above conjecture. We also prove that D1/k is a prime for all sufficiently large integers k.

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Acknowledgements

This work was supported by National Natural Science Foundation of China (Grant Nos. 12171243 and 11922113) and the National Key Research and Development Program of China (Grant No. 2021YFA1000700).

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Authors and Affiliations

  1. School of Mathematical Sciences and Institute of Mathematics, Nanjing Normal University, Nanjing, 210023, China

    Yong-Gao Chen

  2. School of Mathematics and Statistics, Nanjing University of Information Science and Technology, Nanjing, 210044, China

    Quan-Hui Yang

  3. School of Mathematics, Shandong University, Jinan, 250100, China

    Lilu Zhao

Authors
  1. Yong-Gao Chen
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  2. Quan-Hui Yang
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  3. Lilu Zhao
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Corresponding author

Correspondence to Lilu Zhao.

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On the 50th Anniversary of Chen’s Theorem

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Chen, YG., Yang, QH. & Zhao, L. On infinite arithmetic progressions in sumsets. Sci. China Math. 66, 2669–2682 (2023). https://doi.org/10.1007/s11425-023-2177-2

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  • DOI: https://doi.org/10.1007/s11425-023-2177-2

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