## Abstract

A set of positive integers A ⊂ Z_{>0} is log-sparse if there is an absolute constant C so that for any positive integer x the sequence contains at most C elements in the interval [x, 2x). In this note we study arithmetic progressions in sums of log-sparse subsets of Z_{>0}. We prove that for any log-sparse subsets S_{1}, …, S_{n} of Z_{>0}, the sumset S = S_{1} + · · · + S_{n} cannot contain an arithmetic progression of size greater than n^{(1+o(1))n}. We also show that this is nearly tight by proving that there exist log-sparse sets S_{1}, …, S_{n} such that S_{1} +· · · +S_{n} contains an arithmetic progression of size n^{(1−o(1))n}.

Original language | English (US) |
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Article number | A3 |

Journal | Integers |

Volume | 21A |

State | Published - 2021 |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics