2018/10/26 by Jordan Emme, Emme, Jordan, Pascal Hubert +1
Mathematics · #Analytic Number Theory Research #Benford’s Law and Fraud Detection #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Probability (math.PR)
paper · pdf · doi:10.48550/arxiv.1810.11234
openalex publication_date 2018/10/26 · openalex created_date 2018/11/02 · openalex updated_date 2026/07/28
In this paper we study correlation measures introduced in \citeemmeasymptotic2017. Denote by μa(d) the asymptotic density of the set Ea,d=\n ∈ ℕ, s2(n+a)-s2(n)=d\ (where s2 is the sum-of-digits function in base 2). Then, for any point X in \0,1\^ℕ, define the integer sequence (aX (n))n∈ ℕ such that the binary decomposition of aX (n) is the prefix of length n of X. We prove that for any shift-invariant ergodic probability measure ν on \0,1\^ℕ, the sequence (μaX(n))n ∈ ℕ satisfies a central limit theorem. This result was proven in the case where ν is the symmetric Bernoulli measure in \citeemmecentral2018.