On the size of A+λA for algebraic λ

Dmitry Krachun, Fedor Petrov

Research output: Contribution to journalArticlepeer-review


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(Rd) and a compact set K ⊂ Rd 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 languageEnglish (US)
Pages (from-to)117-126
Number of pages10
JournalMoscow Journal of Combinatorics and Number Theory
Issue number2
StatePublished - 2023
Externally publishedYes

All Science Journal Classification (ASJC) codes

  • Algebra and Number Theory
  • Discrete Mathematics and Combinatorics


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


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