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Ergodic theorems with arithmetical weights

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Abstract

We prove that the divisor function d(n) counting the number of divisors of the integer n is a good weighting function for the pointwise ergodic theorem. For any measurable dynamical system (X, A, ν, τ) and any f ∈ L p(ν), p > 1, the limit

$$\mathop {\lim }\limits_{n \to \infty } \frac{1}{{\Sigma _{k = 1}^nd\left( k \right)}}\sum\limits_{k = 1}^n {d\left( k \right)f\left( {{\tau ^k}x} \right)} $$

exists ν-almost everywhere. The proof is based on Bourgain’s method, namely the circle method based on the shift model. Using more elementary ideas we also obtain similar results for other arithmetical functions, like the θ(n) function counting the number of squarefree divisors of n and the generalized Euler totient function J s (n) = Σ d|n d s μ(n/d), s > 0.

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Authors and Affiliations

  1. Ecole Centrale, Grande Voie des Vignes, 92290, Châtenay-Malabry, France

    Christophe Cuny

  2. IRMA, 10 rue du Général Zimmer, 67084, Strasbourg Cedex, France

    Michel Weber

Authors
  1. Christophe Cuny
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  2. Michel Weber
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Corresponding author

Correspondence to Michel Weber.

Additional information

This work was partly carried out during the first author’s visit to Ben-Gurion University, supported by its Center for Advanced Studies in Mathematics, and also during his invitation at the IRMA Institute, Strasbourg, and during the second author’s visit at the Department of Mathematical Sciences, Trondheim, Norway.

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Cuny, C., Weber, M. Ergodic theorems with arithmetical weights. Isr. J. Math. 217, 139–180 (2017). https://doi.org/10.1007/s11856-017-1441-y

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  • DOI: https://doi.org/10.1007/s11856-017-1441-y

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