Ergodic averages with prime divisor weights in L¹

2016/10/03 by Zoltán Buczolich, Buczolich, Zoltan
Mathematics · #28D05 #37A05 #Analytic Number Theory Research #Classical Analysis and ODEs (math.CA) #Dynamical Systems (math.DS) #FOS: Mathematics #Mathematical Dynamics and Fractals #Number Theory (math.NT) #Primary: 37A30 Secondary:11A25

paper · doi:10.48550/arxiv.1610.00511

openalex publication_date 2016/10/03 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We show that ω(n) and Ω(n), the number of distinct prime factors of n and the number of distinct prime factors of n counted according to multiplicity are good weighting functions for the pointwise ergodic theorem in L¹. That is, if g denotes one of these functions and S_g,K=∑_{n≤ K}g(n) then for every ergodic dynamical system (X, \cal A ,μ, τ) and every f∈ L¹(X) lim_K→ ∞ \frac1S_g,K∑_n=1^K g(n)f( τⁿx)=∫_Xfdμ for μ a.e. x∈ X. This answers a question raised by C. Cuny and M. Weber who showed this result for L^p, p>1.

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