Abstract:

Let $A$ be a subset in an abelian group $G$ such that $|A+A| < |A|^{1.99}$. In this case we say that $A$ has moderate doubling. What can be said about the structure of $A$? Essentially nothing is known about this problem. If instead $1.99$ we put $1.01$, i.e. consider sets $A$ with small doubling, then the Polynomial Freiman-Ruzsa conjecture predicts that $A$ has to be close to a subgroup (or Bohr set).

We discuss examples and possible conjectures about the structure of moderate doubling sets. We prove that in the case of $G = F_2^n$, the set $A$ must intersect significantly with a subgroup coset. We follow the entropic approach which was used in the recent resolution of the PFR conjecture over $F_2$.

Joint work with Alex Cohen.