Additive Energy and Irregularities of Distribution

Christoph Aistleitner; Gerhard Larcher

doi:10.1515/udt-2017-0006

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Additive Energy and Irregularities of Distribution

Uniform distribution theory

Volume 12 (2017): Issue 1 (June 2017)

By: Christoph Aistleitner and Gerhard Larcher

Open Access

|Jul 2017

Abstract

We consider strictly increasing sequences (a_n)_n≥1 of integers and sequences of fractional parts ({a_nα})_n≥1 where α ∈ R. We show that a small additive energy of (a_n)_n≥1 implies that for almost all α the sequence ({a_nα})_n≥1 has large discrepancy. We prove a general result, provide various examples, and show that the converse assertion is not necessarily true.

References

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Articles in this issue

DOI: https://doi.org/10.1515/udt-2017-0006 | Journal eISSN: 2309-5377 | Journal ISSN: 1336-913X

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Language: English

Page range: 99 - 107

Submitted on: Mar 7, 2016

Accepted on: Mar 17, 2016

Published on: Jul 22, 2017

Published by: Slovak Academy of Sciences, Mathematical Institute

In partnership with: Paradigm Publishing Services

Publication frequency: 2 issues per year

Keywords:

discrepancy,

additive energy,

metric number theory,

additive combinatorics

Related subjects:

Mathematics,

Algebra and number theory,

Discrete mathematics,

Probability and statistics,

Numerical and computational mathematics

© 2017 Christoph Aistleitner, Gerhard Larcher, published by Slovak Academy of Sciences, Mathematical Institute
This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Volume 12 (2017): Issue 1 (June 2017)