The Bohr radius of the n -dimensional polydisk is equivalent to (log⁡n)/n

We use a new interpolative approach to study the classical polynomial Bohnenblust–Hille inequality. This allows us to greatly improve the best known bounds of this inequality since we actually show that it is subexponential. With this new estimation at hand, we are also able to prove that the Bohr radius of the polydisk DnDn behaves, asymptotically, as log⁡nn. This, in particular, solves several questions from [10].