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We show that the Bohr radius of the polydisk $\mathbb D^n$ behavesasymptotically as $\sqrt{(\log n)/n}$. Our argument is based on a newinterpolative approach to the Bohnenblust--Hille inequalities which allows usto prove that the polynomial Bohnenblust--Hille inequality is subexponential.

The Bohr radius of the $n$-dimensional polydisk is equivalent to $\sqrt{\frac{\log n}{n}}$