On exponential densities and limit ratios of subsets of N

J. Li; L. Olsen

doi:10.1007/s00009-020-01559-7

On exponential densities and limit ratios of subsets of N

J. Li
, L. Olsen

Pure Mathematics

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

Abstract

Given α,β,γ∈[0,1] with α≤β, we prove that there exists a subset of N such that its lower and upper exponential densities and its lower and upper limit ratios are equal to α, β, γ and 1, respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011).

Original language	English
Article number	122
Number of pages	17
Journal	Mediterranean Journal of Mathematics
Volume	17
DOIs	https://doi.org/10.1007/s00009-020-01559-7
Publication status	Published - 28 Jun 2020

Keywords

Positive integer sequence
Exponential densities
Limit ratios

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10.1007/s00009-020-01559-7Licence: CC BY

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Copyright © The Author(s) 2020. Open Access. This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
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title = "On exponential densities and limit ratios of subsets of N",

abstract = "Given α,β,γ∈[0,1] with α≤β, we prove that there exists a subset of N such that its lower and upper exponential densities and its lower and upper limit ratios are equal to α, β, γ and 1, respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011).",

keywords = "Positive integer sequence, Exponential densities, Limit ratios",

author = "J. Li and L. Olsen",

note = "Funding: National Natural Science Foundation of China (11671189,11971109).",

year = "2020",

month = jun,

day = "28",

doi = "10.1007/s00009-020-01559-7",

language = "English",

volume = "17",

journal = "Mediterranean Journal of Mathematics",

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T1 - On exponential densities and limit ratios of subsets of N

AU - Li, J.

AU - Olsen, L.

N1 - Funding: National Natural Science Foundation of China (11671189,11971109).

PY - 2020/6/28

Y1 - 2020/6/28

N2 - Given α,β,γ∈[0,1] with α≤β, we prove that there exists a subset of N such that its lower and upper exponential densities and its lower and upper limit ratios are equal to α, β, γ and 1, respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011).

AB - Given α,β,γ∈[0,1] with α≤β, we prove that there exists a subset of N such that its lower and upper exponential densities and its lower and upper limit ratios are equal to α, β, γ and 1, respectively. This result provides an affirmative answer to an open problem posed by Grekos et al. (Unif Distrib Theory 6:117–130, 2011).

KW - Positive integer sequence

KW - Exponential densities

KW - Limit ratios

U2 - 10.1007/s00009-020-01559-7

DO - 10.1007/s00009-020-01559-7

M3 - Article

SN - 1660-5446

VL - 17

JO - Mediterranean Journal of Mathematics

JF - Mediterranean Journal of Mathematics

M1 - 122

ER -

On exponential densities and limit ratios of subsets of N

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