## Abstract

We prove that for d > 0 and k > 2, for any subset A of a discrete cube {0,1}^{d}, the k—higher energy of A (i.e., the number of 2k-tuples (a1,a2,…, a2k) in A^{2k} with a1 - a2 = a3 - a4 = · · · = a2k—1 — a2k) is at most |A|^{log2(2k+2)}, and log_{2} (2^{k} + 2) is the best possible exponent. We also show that if d > 0 and 2 < k < 10, for any subset A of a discrete cube {0,1}^{d}, the k—additive energy of A (i.e., the number of 2k—tuples (a1,a2,…, a2k) in A^{2k} with a1 +a2 + · · · +ak = ak+1 + ak+2 + · · · +a2k) is at most |A|^{log2 ((Formula presented))}, and log_{2} ((Formula presented)) is the best possible exponent. We discuss the analogous problems for the sets {0, 1,…, n}^{d} for n > 2.

Original language | English |
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Article number | 13 |

Journal | Discrete Analysis |

Volume | 2023 |

DOIs | |

State | Published - 2023 |

Externally published | Yes |

### Bibliographical note

Publisher Copyright:© 2023 J. de Dios Pont, R. Greenfeld, P. Ivanisvili and J. Madrid. All Rights Reserved.

### Funding

We are grateful to the anonymous referees and to the editors of Discrete Analysis for thoughtful comments and corrections which improved the exposition of the paper. We are also thankful to Terence Tao for helpful discussions. This work was initiated at the Hausdorff Research Institute for Mathematics, during the trimester program “Harmonic Analysis and Analytic Number Theory”; we are grateful to the institute and the organizers of the program. *Partially supported by the Eric and Wendy Schmidt Postdoctoral Award, the AMIAS Membership and NSF grants DMS-2242871 and DMS-1926686. †Partially supported by the NSF grants DMS-2152346 and CAREER-DMS-215240.

Funders | Funder number |
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AMIAS | |

National Science Foundation | CAREER-DMS-215240, DMS-2152346, DMS-2242871, DMS-1926686 |

Hausdorff Research Institute for Mathematics |