Bernoulli convolutions and an intermediate value theorem for entropies ofK-partitions

Abstract

We establish a strong regularity property for the distributions of the random sums Σ±λn, known as “infinite Bernoulli convolutions”: For a.e. λ ∃ (1/2, 1) and any fixed l, the conditional distribution of (wn+1...,wn+l) given the sum Σn=0∞wnλn, tends to the uniform distribution on {±1}l asn → ∞. More precise results, where l grows linearly inn, and extensions to other random sums are also obtained. As a corollary, we show that a Bernoulli measure-preserving system of entropyh hasK-partitions of any prescribed conditional entropy in [0,h]. This answers a question of Rokhlin and Sinai from the 1960’s, for the case of Bernoulli systems.