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On doubling and volume: chains

Volume 186 / 2018

Gregory A. Freiman, Oriol Serra Acta Arithmetica 186 (2018), 37-59 MSC: 11P70, 11B75. DOI: 10.4064/aa170211-8-2 Published online: 12 October 2018


The well-known Freiman–Ruzsa theorem provides a structural description of a set $A$ of integers with $|2A|\le c|A|$ as a subset of a $d$–dimensional arithmetic progression $P$ with $|P|\le c’|A|$, where $d$ and $c’$ depend only on $c$. The estimation of the constants $d$ and $c’$ involved in the statement has been the object of intense research. Freiman conjectured in 2008 a formula for the largest volume of such a set. In this paper we prove the conjecture for a general class of sets called chains.


  • Gregory A. FreimanThe Raymond and Beverly Sackler Faculty
    of Exact Sciences
    School of Mathematical Sciences
    Tel Aviv University
    Tel Aviv, Israel
  • Oriol SerraDepartment of Mathematics
    Universitat Politècnica de Catalunya
    Barcelona Graduate School of Mathematics
    Barcelona, Spain

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