A structure theorem for level sets of multiplicative functions and applications

Vitaly Bergelson; Joanna Kułaga-Przymus; Mariusz Lemanczyk; Florian K. Richter

doi:10.1093/IMRN/RNY040

A structure theorem for level sets of multiplicative functions and applications

Vitaly Bergelson, Joanna Kułaga-Przymus, Mariusz Lemanczyk^*, Florian K. Richter

^*Corresponding author for this work

Mathematics

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

Given a level set E of an arbitrary multiplicative function f, we establish, by building on the fundamental work of Frantzikinakis and Host [14, 15], a structure theorem that gives a decomposition of 1E into an almost periodic and a pseudo-random part. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. (Formula Presented). has positive lower density.

Original language	English (US)
Pages (from-to)	1300-1345
Number of pages	46
Journal	International Mathematics Research Notices
Volume	2020
Issue number	5
DOIs	https://doi.org/10.1093/IMRN/RNY040
State	Published - 2020

Funding

This work was supported by National Science Foundation (NSF) [DMS-1500575] to V.B.; Narodowe Centrum Nauki [UMO-2014/15/B/ST1/03736] to J.K.-P. and M.L.; European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [No 647133 (ICHAOS)] to J.K.-P. and the European Union’s [“AOS”, FP7-PEOPLE-2012-IRSES, No 318910] to M.L.

ASJC Scopus subject areas

General Mathematics

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10.1093/IMRN/RNY040

Cite this

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title = "A structure theorem for level sets of multiplicative functions and applications",

abstract = "Given a level set E of an arbitrary multiplicative function f, we establish, by building on the fundamental work of Frantzikinakis and Host [14, 15], a structure theorem that gives a decomposition of 1E into an almost periodic and a pseudo-random part. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. (Formula Presented). has positive lower density.",

author = "Vitaly Bergelson and Joanna Ku{\l}aga-Przymus and Mariusz Lemanczyk and Richter, {Florian K.}",

note = "Funding Information: This work was supported by National Science Foundation (NSF) [DMS-1500575] to V.B.; Narodowe Centrum Nauki [UMO-2014/15/B/ST1/03736] to J.K.-P. and M.L.; European Research Council (ERC) under the European Union{\textquoteright}s Horizon 2020 research and innovation programme [No 647133 (ICHAOS)] to J.K.-P. and the European Union{\textquoteright}s [“AOS”, FP7-PEOPLE-2012-IRSES, No 318910] to M.L. Publisher Copyright: {\textcopyright} 2020 Oxford University Press. All rights reserved.",

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doi = "10.1093/IMRN/RNY040",

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AU - Kułaga-Przymus, Joanna

AU - Lemanczyk, Mariusz

AU - Richter, Florian K.

N1 - Funding Information: This work was supported by National Science Foundation (NSF) [DMS-1500575] to V.B.; Narodowe Centrum Nauki [UMO-2014/15/B/ST1/03736] to J.K.-P. and M.L.; European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme [No 647133 (ICHAOS)] to J.K.-P. and the European Union’s [“AOS”, FP7-PEOPLE-2012-IRSES, No 318910] to M.L. Publisher Copyright: © 2020 Oxford University Press. All rights reserved.

PY - 2020

Y1 - 2020

N2 - Given a level set E of an arbitrary multiplicative function f, we establish, by building on the fundamental work of Frantzikinakis and Host [14, 15], a structure theorem that gives a decomposition of 1E into an almost periodic and a pseudo-random part. Using this structure theorem together with the technique developed by the authors in [3], we obtain the following result pertaining to polynomial multiple recurrence. (Formula Presented). has positive lower density.

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A structure theorem for level sets of multiplicative functions and applications

Abstract

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