## Abstract

For a finite set A ⊂ R and real λ, let A+λA:={a+λb:a, b ∈ A}. We formulate a conjecture about the value of lim inf |A + λA|/|A| for an arbitrary algebraic λ. We support this conjecture by proving a tight lower bound on the Lebesgue measure of K +T K for a given linear operator T ∈ End(R^{d}) and a compact set K ⊂ R^{d} with fixed measure. This continuous result also yields an upper bound in the conjecture. Combining a structural theorem of Freiman on sets with small doubling constants together with a novel discrete analogue of the Prékopa–Leindler inequality we prove a lower bound |A +^{√}2A| ⩾ (1 +^{√}2)^{2} |A| − O(|A|^{1−ε}), which is essentially tight. This proves the conjecture for the specific case λ =^{√}2.

Original language | English (US) |
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Pages (from-to) | 117-126 |

Number of pages | 10 |

Journal | Moscow Journal of Combinatorics and Number Theory |

Volume | 12 |

Issue number | 2 |

DOIs | |

State | Published - 2023 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Discrete Mathematics and Combinatorics

## Keywords

- Leindler inequality
- Minkowski theory, Prékopa
- sumset, Brunn