An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications
| Autor: | Inger J. Håland Knutson, Vitaly Bergelson, Younghwan Son |
|---|---|
| Jazyk: | angličtina |
| Rok vydání: | 2020 |
| Předmět: |
Sequence
Mathematics::Number Theory General Mathematics 010102 general mathematics Vinogradov Zero (complex analysis) Extension (predicate logic) Equidistribution theorem Lambda 01 natural sciences VDP::Matematikk og Naturvitenskap: 400::Matematikk: 410 Combinatorics Integer 0103 physical sciences Multiplication 010307 mathematical physics 0101 mathematics Mathematics |
| Zdroj: | 14965-15018 International Mathematics Research Notices |
| ISSN: | 4965-1501 |
| Popis: | Author's accepted manuscript. This is a pre-copyedited, author-produced version of an article accepted for publication in International Mathematics Research Notices following peer review. The version of record Bergelson, V., Knutson, I. J. H. & Son, Y. (2020). An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications. International Mathematics Research Notices, 2021(19), 14965-15018 is available online at: https://academic.oup.com/imrn/article/2021/19/14965/5775499 and https://doi.org/10.1093/imrn/rnaa035. Generalized polynomials are mappings obtained from the conventional polynomials by the use of the operations of addition and multiplication and taking the integer part. Extending the classical theorem of Weyl on equidistribution of polynomials, we show that a generalized polynomial q(n) has the property that the sequence (q(n)λ)n∈Z is well-distributed mod1 for all but countably many λ∈R if and only if lim|n|→∞n∉J|q(n)|=∞ for some (possibly empty) set J having zero natural density in Z. We also prove a version of this theorem along the primes (which may be viewed as an extension of classical results of Vinogradov and Rhin). Finally, we utilize these results to obtain new examples of sets of recurrence and van der Corput sets. |
| Databáze: | OpenAIRE |
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