Annals of Mathematics· 2012Q1
The quantitative behaviour of polynomial orbits on nilmanifolds
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- Q1SCImago
- 2012year
Short summary
This paper provides a quantitative description of how polynomial orbits on nilmanifolds become uniformly distributed, refining Leibman's theorem.
AI-generated from the title and abstract; the full text is not read.
Key points
- Polynomial orbits on nilmanifolds are quantitatively described in their uniform distribution.
- A factorization of polynomial orbits into 'smooth', 'periodic/rational', and uniformly distributed parts is established.
- The uniform distribution is shown to occur within a subnilmanifold up to a specified error.
- Bounds on this distribution are uniform in N and polynomial in the error tolerance δ.
AI-generated from the title and abstract; the full text is not read.
Abstract
A theorem of Leibman asserts that a polynomial orbit (g(n)Γ) n∈Z on a nilmanifold G/Γ is always equidistributed in a union of closed subnilmanifolds of G/Γ.In this paper we give a quantitative version of Leibman's result, describing the uniform distribution properties of a finite polynomial orbit (g(n)Γ) n∈[N ] in a nilmanifold.More specifically we show that there is a factorisation g = εg γ, where ε(n) is "smooth," (γ(n)Γ) n∈Z is periodic and "rational," and (g (n)Γ)n∈P is uniformly distributed (up to a specified error δ) inside some subnilmanifold G /Γ of G/Γ for all sufficiently dense arithmetic progressions P ⊆ [N ].Our bounds are uniform in N and are polynomial in the error tolerance δ.
The authors' abstract, as published at the source. Annals of Mathematics, 2012 · DOI ↗
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Field: Discrete Mathematics and Combinatorics
Discrete Mathematics and CombinatoricsMathematics