Bernoulli convolutions and an intermediate value theorem for entropies of K -partitions

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