Nilsequences and Multiple Correlations along Subsequences

TL;DR

This paper refines the decomposition of multiple polynomial correlation sequences into nilsequences and null sequences, showing the null part vanishes along primes and Hardy sequences, with counterexamples for rigid sequences.

Contribution

It improves existing results by proving null sequences vanish along primes and Hardy sequences and constructs counterexamples for rigid sequences.

Findings

Null sequences go to zero in density along primes and Hardy sequences.

Decomposition of correlation sequences into nilsequences and null sequences is refined.

Counterexamples show null sequences may not vanish along certain rigid sequences.

Abstract

The results of Bergelson-Host-Kra and Leibman say that a multiple polynomial correlation sequence can be decomposed into a sum of a nilsequence (a sequence defined by evaluating a continuous function along an orbit in a nilsystem) and a null sequence (a sequence that goes to zero in density). We refine their results by proving that the null sequence goes to zero in density along polynomials evaluated at primes and Hardy sequence $(\lfloor n^c \rfloor)$. On the other hand, given a rigid sequence, we construct an example of correlation whose null sequence does not approach zero in density along that rigid sequence. As a corollary of a lemma in the proof, the formula for the pointwise ergodic average along polynomials of primes in a nilsystem is also obtained.