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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Orbit of the diagonal in the power of a nilmanifold


Author: A. Leibman
Journal: Trans. Amer. Math. Soc. 362 (2010), 1619-1658
MSC (2000): Primary 37C99; Secondary 22E25
Published electronically: October 20, 2009
MathSciNet review: 2563743
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Abstract: Let $ X$ be a nilmanifold, that is, a compact homogeneous space of a nilpotent Lie group $ G$, and let $ a\in G$. We study the closure of the orbit of the diagonal of $ X^{r}$ under the action $ (a^{p_{1}(n)} ,\ldots ,a^{p_{r}(n)})$, where $ p_{i}$ are integer-valued polynomials in $ m$ integer variables. (Knowing this closure is crucial for finding limits of the form $ \hbox {lim}_{N\rightarrow \infty }\frac{1}{N^{m}}\sum _{n\in \{1,\ldots ,N\}^{m}} \mu (T^{p_{1}(n)}A_{1}\cap \ldots \cap T^{p_{r}(n)}A_{r})$, where $ T$ is a measure-preserving transformation of a finite measure space $ (Y,\mu )$ and $ A_{i}$ are subsets of $ Y$, and limits of the form $ \hbox {lim}_{N\rightarrow \infty }\frac{1}{N^{m}}\sum _{n\in \{1,\ldots ,N\}^{m}} d((A_{1}+p_{1}(n))\cap \ldots \cap (A_{r}+p_{r}(n)))$, where $ A_{i}$ are subsets of Z and $ d(A)$ is the density of $ A$ in Z.) We give a simple description of the closure of the orbit of the diagonal in the case that all $ p_{i}$ are linear, in the case that $ G$ is connected, and in the case that the identity component of $ G$ is commutative; in the general case our description of the orbit is not explicit.


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Additional Information

A. Leibman
Affiliation: Department of Mathematics, The Ohio State University, Columbus, Ohio 43210
Email: leibman@math.ohio-state.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-09-04961-7
PII: S 0002-9947(09)04961-7
Keywords: Orbits on nilmanifolds, complexity of polynomial systems
Received by editor(s): May 27, 2008
Published electronically: October 20, 2009
Additional Notes: This research was supported in part by NSF grants DMS-0345350 and DMS-0600042.
Article copyright: © Copyright 2009 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.