A decomposition of multicorrelation sequences for commuting transformations along primes

Abstract

We study multicorrelation sequences arising from systems with commuting transformations. Our main result is a refinement of a decomposition result of Frantzikinakis and it states that any multicorrelation sequences for commuting transformations can be decomposed, for every $\epsilon >0$, as the sum of a nilsequence $\varphi (n)$ and a sequence $\omega (n)$ satisfying ${\lim }_{N\to \infty }\frac{1}{N}{\sum }_{n=1}^N|\omega (n)|<\epsilon$ and ${\lim }_{N\to \infty }\frac{1}{|\mathbb{P}\cap [N]|}{\sum }_{p\in \mathbb{P}\cap [N]}|\omega (p)|<\epsilon$.

27 pages

References in corpus (3)

• The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
• Weighted multiple ergodic averages and correlation sequences
• Nilsequences and Multiple Correlations along Subsequences

Cited by in corpus (1)

• Structure of multicorrelation sequences with integer part polynomial iterates along primes