Nilsequences, null-sequences, and multiple correlation sequences

Abstract

A ($d$-parameter) basic nilsequence is a sequence of the form $\psi (n)= f({a}^{n} x)$,$n\in { \mathbb{Z} }^{d} $, where $x$ is a point of a compact nilmanifold $X$, $a$ is a translation on $X$, and$f\in C(X)$; a nilsequence is a uniform limit of basic nilsequences. If $X= G/ \Gamma $ is a compact nilmanifold, $Y$ is a subnilmanifold of $X$, $\mathop{(g(n))}\nolimits_{n\in { \mathbb{Z} }^{d} } $ is a polynomial sequence in $G$, and $f\in C(X)$, we show that the sequence $\phi (n)= \int \nolimits \nolimits_{g(n)Y} f$ 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 $(W, \mathcal{B} , \mu , T)$, polynomials ${p}_{1} , \ldots , {p}_{k} : { \mathbb{Z} }^{d} \longrightarrow \mathbb{Z} $, and sets ${A}_{1} , \ldots , {A}_{k} \in \mathcal{B} $, the sequence $\phi (n)= \mu ({T}^{{p}_{1} (n)} {A}_{1} \cap \cdots \cap {T}^{{p}_{k} (n)} {A}_{k} )$, $n\in { \mathbb{Z} }^{d} $, is the sum of a nilsequence and a null-sequence.