7 - Probabilistic Tools I

Summary

We give an alternative, probabilistic, approach to two of the subjects considered so far: the optimality of the exponent in the polynomial Bohnenblust-Hille inequality (see Chapter 6) and the lower bound for S in Bohr’s problem (see Chapters 1 and 4). We use a probabilistic device: the Kahane-Salem-Zygmund inequality. This shows that, for a given finite family of coefficients, a choice of signs can be found in such a way that the polynomial whose coefficients are the original ones multiplied by the signs has small norm (supremum on the polydisc). The proof uses Bernstein’s inequality and Rademacher random variables. We also look at the relationship between Rademacher and Steinhaus random variables, and deduce the classical Khinchin inequality from the Khinchin-Steinhaus inequality (see Chapter 6). We consider Dirichlet series, place signs before the coefficients, and define the almost sure abscissas (in each of the senses from Chapter 1) by considering each convergence for almost every choice of signs. An analogue of Bohr’s problem in this sense is considered.