Seminar 11.1.18 Le | Ergodic Theory at Ohio State

Title: Subsequences of multiple correlations

Speaker: Anh Le (Northwestern University)

Abstract: The results of Bergelson-Host-Kra and of Leibman say that a multiple correlation sequence can be decomposed as a sum of a nilsequence and a null sequence. Inspired by these results, Frantzikinakis asks the following question: Let $r_{n}$ be the sequence of primes, or $[n^{c}]$ , or $2^{n}$ . For a multiple correlation $a (n)$ , is it true that there exists a nilsequence $b (n)$ and null sequence $e (n)$ such that $a (r_{n}) = b (r_{n}) + e (n)$ ?

In this talk, I’ll briefly discuss why the answer is affirmative for the sequences of primes and $[n^{c}]$ . However, our main focus will be about $2^{n}$ . The answer for this sequence is also yes and surprisingly related to the notion of sets of Bohr recurrence.