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

Abstract

Abstract. We show that the Bohr radius of the polydisk D n behaves asymptotically as (log n)/n. Our argument is based on a new interpolative approach to the Bohnenblust-Hille inequalities which allows us to prove, among other results, that the polynomial Bohnenblust-Hille inequality is subexponential.

Isomorphic copies of ℓ1 for m‐homogeneous non‐analytic Bohnenblust–Hille polynomials

“…Recently, the work by Bohnenblust and Hille on Dirichlet series has resurfaced and attracted the attention of many authors who became interested in classical problems such as obtaining the optimal values for the constants in the Bohnenblust–Hille and Hardy–Littlewood inequalities, or in estimating the asymptotic value of the n ‐dimensional Bohr radius (see, e.g., , , ).…”

“…Moreover, we can have this copy of 1 in such a way that every non-zero element of it fails to be analytic at precisely the same point.The existence of elements for which the convergence at a certain point fails is not a isolated phenomenon. Bohnenblust and Hille in [5] solved a long standing problem on Dirichlet series and, as one of the tools for the solution, extended this construction of Toeplitz to m-homogeneous polynomials (see Proposition 3.1).Recently, the work by Bohnenblust and Hille [5] on Dirichlet series has resurfaced and attracted the attention of many authors who became interested in classical problems such as obtaining the optimal values for the constants in the Bohnenblust-Hille and Hardy-Littlewood inequalities, or in estimating the asymptotic value of the n-dimensional Bohr radius (see, e.g., [3], [6], [9]).…”

Bohr radius for the punctured disk

“… obtained the optimal asymptotic estimate for this radius by using the fact that the Bohnenblust–Hille inequality is indeed hypercontractive. The exact asymptotic behaviour of the radius was obtained by Bayart, Pellegrino and Seoane‐Sepúlveda .…”

On unimodular multilinear forms with small norms on sequence spaces

“…Let K be the real or complex scalar field. The Kahane-Salem-Zygmund inequality (see [3,4]) asserts that for positive integers m, n and p 1 , ..., p m ∈ [2, ∞], there exist a universal constant C (depending only on m), a choice of signs 1 and −1, and an m-linear form A m,n : ℓ n p 1 ×· · ·×ℓ n pm −→ K of the type A m,n (z (1) , ..., z (m) ) = n j 1 ,...,jm=1 ±z (1)…”