On doubling and volume: chains
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.