Arithmetic progressions -- an operator theoretic view
Tanja Eisner Rainer Nagel
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
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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