  We establish new bounds in the Bogolyubov-Ruzsa lemma, demonstrating that if
A is a subset of a finite abelian group with density alpha, then 3A-3A contains
a Bohr set of rank O(log^2 (2/alpha)) and radius Omega(log^{-2} (2/alpha)). The
Bogolyubov-Ruzsa lemma is one of the deepest results in additive combinatorics,
with a plethora of important consequences. In particular, we obtain new results
toward the Polynomial Freiman-Ruzsa conjecture and improved bounds in Freiman's
theorem.
