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On the size of $A{+}\lambda A$ for algebraic $\lambda$

Dmitry Krachun and Fedor Petrov

Vol. 12 (2023), No. 2, 117–126
Abstract

For a finite set A and real λ, let A + λA := {a + λb : a,b A}. We formulate a conjecture about the value of liminf |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 (d) and a compact set K 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|1𝜀), which is essentially tight. This proves the conjecture for the specific case λ = 2.

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Keywords
sumset, Brunn–Minkowski theory, Prékopa–Leindler inequality
Mathematical Subject Classification
Primary: 05B10
Secondary: 11B13
Milestones
Received: 28 July 2022
Revised: 15 November 2022
Accepted: 1 December 2022
Published: 4 June 2023
Authors
Dmitry Krachun
Department of Mathematics and Computer Science
St. Petersburg State University
St. Petersburg
Russia
Fedor Petrov
Department of Mathematics and Computer Science
St. Petersburg State University
St. Petersburg
Russia