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Nilsequences, null-sequences, and multiple correlation sequences

  • A. LEIBMAN (a1)

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.

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[BHK]Bergelson, V., Host, B. and Kra, B.. Multiple recurrence and nilsequences. Invent. Math. 160 (2) (2005), 261303.
[BL]Bergelson, V. and Leibman, A.. Distribution of values of bounded generalized polynomials. Acta Math. 198 (2007), 155230.
[GT]Green, B. and Tao, T.. The quantitative behaviour of polynomial orbits on nilmanifolds. Ann. of Math. (2) 175 (2) (2012), 465540.
[Ha1]Håland, I. J.. Uniform distribution of generalized polynomials. J. Number Theory 45 (1993), 327366.
[Ha2]Håland, I. J.. Uniform distribution of generalized polynomials of the product type. Acta Arith. 67 (1994), 1327.
[HK1]Host, B. and Kra, B.. Non-conventional ergodic averages and nilmanifolds. Ann. of Math. (2) 161 (1) (2005), 397488.
[HK2]Host, B. and Kra, B.. Nil-Bohr sets of integers. Ergod. Th. & Dynam. Sys. 31 (2011), 113142.
[HKM]Host, B., Kra, B. and Maass, A.. Nilsequences and a structure theorem for topological dynamical systems. Adv. Math. 224 (2010), 103129.
[L1]Leibman, A.. Pointwise convergence of ergodic averages for polynomial sequences of translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 201213.
[L2]Leibman, A.. Pointwise convergence of ergodic averages for polynomial actions of ${ \mathbb{Z} }^{d} $ by translations on a nilmanifold. Ergod. Th. & Dynam. Sys. 25 (2005), 215225.
[L3]Leibman, A.. Convergence of multiple ergodic averages along polynomials of several variables. Israel J. Math. 146 (2005), 303315.
[L4]Leibman, A.. Rational sub-nilmanifolds of a compact nilmanifold. Ergod. Th. & Dynam. Sys. 26 (2006), 787798.
[L5]Leibman, A.. Orbit of the diagonal in the power of a nilmanifold. Trans. Amer. Math. Soc. 362 (2010), 16191658.
[L6]Leibman, A.. Multiple polynomial correlation sequences and nilsequences. Ergod. Th. & Dynam. Sys. 30 (2010), 841854.
[L7]Leibman, A.. A canonical form and the distribution of values of generalized polynomials. Israel J. Math. 188 (2012), 131176.
[M]Malcev, A.. On a class of homogeneous spaces. Amer. Math. Soc. Transl. 9 (1962), 276307.
[Z]Ziegler, T.. Universal characteristic factors and Furstenberg averages. J. Amer. Math. Soc. 20 (1) (2007), 5397.

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Nilsequences, null-sequences, and multiple correlation sequences

  • A. LEIBMAN (a1)

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