Polynomial multiple recurrence and large intersections in rings of integers

15 Jul 2021  ·  Ethan Ackelsberg, Vitaly Bergelson ·

We establish characteristic factors for natural classes of polynomial multiple ergodic averages in rings of integers and derive corresponding Khintchine-type recurrence theorems, extending results of Frantzikinakis and Kra and of Frantzikinakis about polynomial configurations in $\mathbb{Z}$. Using previous work of Griesmer and of the second author and Robertson, we reduce the problem of finding characteristic factors to proving a result on equidistribution of polynomial orbits in nilmanifolds, which is of independent interest...

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Dynamical Systems Combinatorics 37A44 (Primary) 05D10, 37A15, 37A30 (Secondary)