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arXiv · 0808.0142

Ergodic averages with deterministic weights

Fabien Durand·Dominique Schneider
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Abstract

The purpose of this paper is to study ergodic averages with deterministic weights. More precisely we study the convergence of the ergodic averages of the type $\frac{1}{N} \sum_{k=0}^{N-1} θ(k) f \circ T^{u_k}$ where $θ= (θ(k) ; k\in \NN)$ is a bounded sequence and $u = (u_k ; k\in \NN)$ a strictly increasing sequence of integers such that for some $δ<1$ $$ S_N (θ, u) := \sup_{α\in \pRR} | \sum_{k=0}^{N-1} θ(k) \exp (2iπαu_k) | = O (N^δ) \leqno{({\cal H}_1)} $$ i.e., there exists a constant $C$ such that $S_N (θ, u) \leq C N^δ $. We define $δ(θ, u)$ to be the infimum of the $δ$ satisfying $\H_1$ for $θ$ and $u$.

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Fabien Durand, Dominique Schneider. 2008-08-01. Ergodic averages with deterministic weights. https://arxiv.org/abs/0808.0142

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