## 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 logsparse 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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Title of host publication | Number Theory and Combinatorics |

Subtitle of host publication | A Collection in Honor of the Mathematics of Ronald Graham |

Publisher | de Gruyter |

Pages | 27-33 |

Number of pages | 7 |

ISBN (Electronic) | 9783110754216 |

ISBN (Print) | 9783110753431 |

DOIs | |

State | Published - Apr 19 2022 |

## All Science Journal Classification (ASJC) codes

- General Mathematics