Nilsequences, null-sequences, and multiple correlation sequences

Abstract

A (d-parameter) basic nilsequence is a sequence of the form ψ(n)=f(a^{n}x), n \in Z^{d}, where x is a point of a compact nilmanifold X, a is a translation on X, and f is a continuous function on X; a nilsequence is a uniform limit of basic nilsequences. If X is a compact nilmanifold, Y is a subnilmanifold of X, g(n) is a (d-parameter) polynomial sequence of translations of X, and f is a continuous function on X, we show that the sequence \int_{g(n)Y}f is the sum of a basic nilsequence and a sequence that converges to zero in uniform density (a null-sequence). We also show that an integral of a family of nilsequences is a nilsequence plus a null-sequence. We deduce that for any invertible finite measure preserving system (W,μ,T), integer polynomials p_{1},...,p_{k} on Z^{d}, and measurable sets A_{1},...,A_{k} in W, the sequence μ(T^{p_{1}(n)}A_{1}\cap...\cap T^{p_{k}(n)}A_{k}), n\in Z^{d}, is the sum of a nilsequence and a null-sequence.

Cited by in corpus (14)

• The structure of logarithmically averaged correlations of multiplicative functions, with applications to the Chowla and Elliott conjectures
• Some open problems on multiple ergodic averages
• Multiple correlation sequences and nilsequences
• 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
• Integer part polynomial correlation sequences
• Weighted multiple ergodic averages and correlation sequences
• Nilsequences and Multiple Correlations along Subsequences
• Nilsystems and ergodic averages along primes
• Sequences from zero entropy noncommutative toral automorphisms and Sarnak Conjecture
• A decomposition of multicorrelation sequences for commuting transformations along primes
• Structure of multicorrelation sequences with integer part polynomial iterates along primes
• Simultaneous approximation in nilsystems and the multiplicative thickness of return-time sets
• Multiple recurrence and popular differences for polynomial patterns in rings of integers