Automatic sequences as good weights for ergodic theorems
Tanja Eisner Jakub Konieczny
Discrete & Continuous Dynamical Systems - A 2018, 38(8): 4087-4115 doi: 10.3934/dcds.2018178

We study correlation estimates of automatic sequences (that is, sequences computable by finite automata) with polynomial phases. As a consequence, we provide a new class of good weights for classical and polynomial ergodic theorems. We show that automatic sequences are good weights in $ L^2$ for polynomial averages and totally ergodic systems. For totally balanced automatic sequences (i.e., sequences converging to zero in mean along arithmetic progressions) the pointwise weighted ergodic theorem in $ L^1$ holds. Moreover, invertible automatic sequences are good weights for the pointwise polynomial ergodic theorem in $ L^r$, $ r>1$.

keywords: Automatic sequence ergodic theorem Wiener–Wintner
Arithmetic progressions -- an operator theoretic view
Tanja Eisner Rainer Nagel
Discrete & Continuous Dynamical Systems - S 2013, 6(3): 657-667 doi: 10.3934/dcdss.2013.6.657
Motivated by the recent Green--Tao theorem on arithmetic progressions in the primes, we discuss some of the basic operator theoretic techniques used in its proof. In particular, we obtain a complete proof of Szemerédi's theorem for arithmetic progressions of length $3$ (Roth's theorem) and the Furstenberg--Sárközy theorem.
keywords: Roth's theorem Jacobs-Glicksberg-deLeeuw decomposition. multiple ergodic theorems Arithmetic progressions operator theoretic methods
Uniformity in the Wiener-Wintner theorem for nilsequences
Tanja Eisner Pavel Zorin-Kranich
Discrete & Continuous Dynamical Systems - A 2013, 33(8): 3497-3516 doi: 10.3934/dcds.2013.33.3497
We prove a uniform extension of the Wiener-Wintner theorem for nilsequences due to Host and Kra and a nilsequence extension of the topological Wiener-Wintner theorem due to Assani. Our argument is based on (vertical) Fourier analysis and a Sobolev embedding theorem.
keywords: Wiener-Wintner theorem nilsequence uniform convergence.

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