Nilsequences and Multiple Correlations along Subsequences

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.

21 pages

References in corpus (3)

• The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
• An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement)
• Nilsystems and ergodic averages along primes

Cited by in corpus (6)

• The structure of correlations of multiplicative functions at almost all scales, with applications to the Chowla and Elliott conjectures
• Odd order cases of the logarithmically averaged Chowla conjecture
• Higher uniformity of arithmetic functions in short intervals I. All intervals
• Structure of multicorrelation sequences with integer part polynomial iterates along primes
• A decomposition of multicorrelation sequences for commuting transformations along primes
• Optimal lower bounds for multiple recurrence