The Bohnenblust-Hille inequality for homogeneous polynomials is hypercontractive

The Bohnenblust–Hille inequality says that the l 2m m+1 -norm of the coefficients of an m-homogeneous polynomial P on C is bounded by ‖P‖∞ times a constant independent of n, where ‖ · ‖∞ denotes the supremum norm on the polydisc D. The main result of this paper is that this inequality is hypercontractive, i.e., the constant can be taken to be C for some C > 1. Combining this improved version of the Bohnenblust–Hille inequality with other results, we obtain the following: The Bohr radius for the polydisc D behaves asymptotically as √ (log n)/n modulo a factor bounded away from 0 and infinity, and the Sidon constant for the set of frequencies { logn : n a positive integer ≤ N } is √ N exp{(−1/ √ 2+o(1)) √ logN log logN} as N → ∞.