Functions tiling simultaneously with two arithmetic progressions

Mark Mordechai Etkind; Nir Lev

doi:10.1112/plms.12570

Functions tiling simultaneously with two arithmetic progressions

Mark Mordechai Etkind
, Nir Lev

Department of Mathematics

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

1 Scopus citations

Abstract

We consider measurable functions (Formula presented.) on (Formula presented.) that tile simultaneously by two arithmetic progressions (Formula presented.) and (Formula presented.) at respective tiling levels (Formula presented.) and (Formula presented.). We are interested in two main questions: what are the possible values of the tiling levels (Formula presented.), and what is the least possible measure of the support of (Formula presented.) ? We obtain sharp results which show that the answers depend on arithmetic properties of (Formula presented.) and (Formula presented.), and in particular, on whether the numbers (Formula presented.) are rationally independent or not.

Original language	English
Pages (from-to)	1775-1815
Number of pages	41
Journal	Proceedings of the London Mathematical Society
Volume	127
Issue number	6
DOIs	https://doi.org/10.1112/plms.12570
State	Published - Dec 2023

Bibliographical note

Publisher Copyright:
© 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.

Funding

We thank Mihalis Kolountzakis for posing to us the problem discussed in Section 6. This research was supported by ISF Grant Number: 1044/21 and ERC Starting Grant Number: 713927. We thank Mihalis Kolountzakis for posing to us the problem discussed in Section 6 . This research was supported by ISF Grant Number: 1044/21 and ERC Starting Grant Number: 713927.

Funders	Funder number
Mihalis Kolountzakis
European Commission	713927
Israel Science Foundation	1044/21

Access to Document

10.1112/plms.12570

Cite this

@article{ad6ae84f9cef42aaa5e61db2305b5179,

title = "Functions tiling simultaneously with two arithmetic progressions",

abstract = "We consider measurable functions (Formula presented.) on (Formula presented.) that tile simultaneously by two arithmetic progressions (Formula presented.) and (Formula presented.) at respective tiling levels (Formula presented.) and (Formula presented.). We are interested in two main questions: what are the possible values of the tiling levels (Formula presented.), and what is the least possible measure of the support of (Formula presented.) ? We obtain sharp results which show that the answers depend on arithmetic properties of (Formula presented.) and (Formula presented.), and in particular, on whether the numbers (Formula presented.) are rationally independent or not.",

author = "Etkind, \{Mark Mordechai\} and Nir Lev",

note = "Publisher Copyright: {\textcopyright} 2023 The Authors. The publishing rights in this article are licensed to the London Mathematical Society under an exclusive licence.",

year = "2023",

month = dec,

doi = "10.1112/plms.12570",

language = "אנגלית",

volume = "127",

pages = "1775--1815",

journal = "Proceedings of the London Mathematical Society",

issn = "0024-6115",

publisher = "John Wiley and Sons Ltd",

number = "6",

}

TY - JOUR

T1 - Functions tiling simultaneously with two arithmetic progressions

AU - Etkind, Mark Mordechai

AU - Lev, Nir

PY - 2023/12

Y1 - 2023/12

N2 - We consider measurable functions (Formula presented.) on (Formula presented.) that tile simultaneously by two arithmetic progressions (Formula presented.) and (Formula presented.) at respective tiling levels (Formula presented.) and (Formula presented.). We are interested in two main questions: what are the possible values of the tiling levels (Formula presented.), and what is the least possible measure of the support of (Formula presented.) ? We obtain sharp results which show that the answers depend on arithmetic properties of (Formula presented.) and (Formula presented.), and in particular, on whether the numbers (Formula presented.) are rationally independent or not.

AB - We consider measurable functions (Formula presented.) on (Formula presented.) that tile simultaneously by two arithmetic progressions (Formula presented.) and (Formula presented.) at respective tiling levels (Formula presented.) and (Formula presented.). We are interested in two main questions: what are the possible values of the tiling levels (Formula presented.), and what is the least possible measure of the support of (Formula presented.) ? We obtain sharp results which show that the answers depend on arithmetic properties of (Formula presented.) and (Formula presented.), and in particular, on whether the numbers (Formula presented.) are rationally independent or not.

UR - https://www.scopus.com/pages/publications/85174563561

U2 - 10.1112/plms.12570

DO - 10.1112/plms.12570

M3 - ???researchoutput.researchoutputtypes.contributiontojournal.article???

AN - SCOPUS:85174563561

SN - 0024-6115

VL - 127

SP - 1775

EP - 1815

JO - Proceedings of the London Mathematical Society

JF - Proceedings of the London Mathematical Society

IS - 6

ER -

Functions tiling simultaneously with two arithmetic progressions

Abstract

Bibliographical note

Funding

Access to Document

Other files and links

Fingerprint

Cite this