6533b827fe1ef96bd1287055

RESEARCH PRODUCT

An Extension of Weyl’s Equidistribution Theorem to Generalized Polynomials and Applications

Inger J. Håland KnutsonVitaly BergelsonYounghwan Son

subject

SequenceMathematics::Number TheoryGeneral Mathematics010102 general mathematicsVinogradovZero (complex analysis)Extension (predicate logic)Equidistribution theoremLambda01 natural sciencesVDP::Matematikk og Naturvitenskap: 400::Matematikk: 410CombinatoricsInteger0103 physical sciencesMultiplication010307 mathematical physics0101 mathematicsMathematics

description

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.

yearjournalcountryeditionlanguage
2020-03-03
10.1093/imrn/rnaa035https://hdl.handle.net/11250/3046608