Some open problems on multiple ergodic averages

Abstract

We survey some recent developments and give a list of open problems regarding multiple recurrence and convergence phenomena of ${\mathbb{Z}}^d$ actions in ergodic theory and related applications in combinatorics and number theory.

Remarks by the referees incorporated. To appear in the Bulletin of the Hellenic Mathematical Society. Updates on the status of the problems will be posted here: http://www.math.uoc.gr/~nikosf/OpenProblems.html

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• Norm convergence of multiple ergodic averages on amenable groups
• Integer part polynomial correlation sequences
• Nilsequences and Multiple Correlations along Subsequences
• Uniform distribution in nilmanifolds along functions from a Hardy field
• Multiple ergodic averages along functions from a Hardy field: convergence, recurrence and combinatorial applications
• $\Delta$-weakly mixing subsets along a collection of sequences of integers
• Structure of multicorrelation sequences with integer part polynomial iterates along primes
• Multiple correlation sequences not approximable by nilsequences
• Convergence of ergodic-martingale paraproducts
• A decomposition of multicorrelation sequences for commuting transformations along primes
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