The well-known Freiman–Ruzsa theorem provides a structural description of a set A of integers with |2A|=c|A| as a subset of a d–dimensional arithmetic progression P with |P|=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.
Freiman, G., Serra, O. On doubling and volume: chains. "Acta Arithmetica", 1 Octubre 2018, vol. 186, núm. 1, p. 37-59.