113 research outputs found
A short proof of a conjecture of Erd\"os proved by Moreira, Richter and Robertson
We give a short proof of a sumset conjecture of Erd\"os, recently proved by
Moreira, Richter and Robertson: every subset of the integers of positive
density contains the sum of two infinite sets. The proof is written in the
framework of classical ergodic theory.Comment: Made some small correction
Ergodic seminorms for commuting transformations and applications
Recently, T. Tao gave a finitary proof a convergence theorem for multiple
averages with several commuting transformations and soon later, T. Austin gave
an ergodic proof of the same result. Although we give here one more proof of
the same theorem, this is not the main goal of this paper. Our main concern is
to provide some tools for the case of several commuting transformations,
similar to the tools successfully used in the case of a single transformation,
with the idea that they will be useful in the solution of other problems
Analysis of two step nilsequences
Nilsequences arose in the study of the multiple ergodic averages associated
to Furstenberg's proof of Szemer\'edi's Theorem and have since played a role in
problems in additive combinatorics. Nilsequences are a generalization of almost
periodic sequences and we study which portions of the classical theory for
almost periodic sequences can be generalized for two step nilsequences. We
state and prove basic properties for 2-step nilsequences and give a
classification scheme for them
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