Distribution of values of bounded generalized polynomials

By Vitaly Bergelson, Alexander Leibman. 2007

97 citations recorded by the source corpus.

A generalized polynomial is a real-valued function which is obtained from conventional polynomials by the use of the operations of addition, multiplication, and taking the integer part; a generalized polynomial mapping is a vector-valued mapping whose coordinates are generalized polynomials. We show that any bounded generalized polynomial mapping u: Zd → Rl has a representation u(n) = f(ϕ(n)x), n ∈ Zd, where f is a piecewise polynomial function on a compact nilmanifold X, x ∈ X, and ϕ is an ergodic Zd-action by translations on X. This fact is used to show that the sequence u(n), n ∈ Zd, is well distributed on a piecewise polynomial surface $\mathcal{S}\subset\mathbf{R}^{l}$ (with respect to the Borel measure on $\mathcal{S}$ that is the image of the Lebesgue measure under the piecewise polynomial function defining $\mathcal{S}$). As corollaries we also obtain a von Neumann-type ergodic theorem along generalized polynomials and a result on Diophantine approximations extending the work of van der Corput and of Furstenberg–Weiss.

Subfields: Applied Mathematics, Geometry and Topology, Mathematical Physics

View Paper: https://www.semanticscholar.org/paper/31be6c11b13ac261cd9bca724e8d2b7c487c69b7

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In Applied Mathematics

In Geometry and Topology

In Mathematical Physics

Citations in the corpus

The Mobius function is strongly orthogonal to nilsequences (2008, background): “Indeed the results of Leibman [12] on the distribution of (g(n)Γ)n∈Z imply that the sequence (F (g(n)Γ))n∈Z cannot have isolated values, yet there are bracket polynomials which do.”
Nil Bohr-sets and almost automorphy of higher order (2014, background): “[6] V. Bergelson and A. Leibman, Distribution of values of bounded generalized polynomials, Acta Math., 198 (2007), no. 2, 155–230.”
An inverse theorem for the Gowers U^{s+1}[N]-norm (announcement) (2010, unknown): “One can describe equivalence of bracket polynomials explicitly using bracket calculus, as developed in [37] (see also the earlier works [4], [24], [25], [26]), but this requires a fair amount of notation and machinery.”
Some open problems on multiple ergodic averages (2011, background): “for α,β∈ R, the sequence (ei[nα]nβ) is a basic generalized 2-step nilsequence deﬁned on the nilmanifold given by the direct product of the nilmanifolds in Examples 1 and 2 above. It can be shown (see [55]) that if αand βare rationally independent, then the sequence (ei[nα]nβ) is asymptotically ortho”
Rigidity and non-recurrence along sequences (2011, methodology): “To cite just two references, see Lemańczyk et al [41] for examples that are constructed randomly, and see Bergelson and Leibman [5] for examples given by generalized polynomials.”