2023/11/01 by Nikita Sidorov, Sidorov, Nikita
Mathematics · #11R06 #28D40 #Advanced Mathematical Identities #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Number Theory (math.NT) #advanced mathematical theories
paper · pdf · doi:10.48550/arxiv.2311.00569
openalex publication_date 2023/11/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Let θ∈(1,2), and μθ be the Bernoulli convolution parametrized by θ, that is, the measure corresponding to the distribution of the random variable ∑n=1∞ anθ-n, where the an are i.i.d. with probability of an=0 equal to \frac12. As is well known, μθ is either equivalent to the Lebesgue measure on supp(μθ), or singular. Recall that an algebraic integer >1 is called Pisot if all its other Galois conjugates are smaller than 1 in modulus. It is known that μθ is singular with dimμθ<1 if θ is Pisot. An algebraic integer θ greater than 1 is called a Salem number if all its other Galois conjugates are of modulus 1, except θ-1. I shall prove that (1) dimμθ=1 if θ is an algebraic non-Pisot number. (2) if θ is Salem, then μθ is equivalent to the Lebesgue measure on supp(μθ), with an unbounded density in Lp(supp(μθ)) for all p<∞. (3) Define βθ,x,n=#\a1… an: ∃ an+1\dotssuch that x=∑k=1∞anθ-k\. Then limn→∞√[n]βθ,x,n=θdimμθ for μθ-a.e. x. (4) Put \[ \bigcupn=1^∞\∑k=1nakθk| ak∈\-1,0,1\\= \y0(θ)