The density of sets containing large similar copies of finite sets

Kenneth Falconer; Vjekoslav Kovač; Alexia Yavicoli

doi:10.1007/s11854-022-0231-6

The density of sets containing large similar copies of finite sets

Kenneth Falconer, Vjekoslav Kovač, Alexia Yavicoli^*

^*Corresponding author for this work

Pure Mathematics

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

Abstract

We prove that if E⊆R^d (d≥2) is a Lebesgue-measurable set with density larger than (n−2) (n−1), then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1−O(n^−1/5log n).

Original language	English
Pages (from-to)	339-359
Number of pages	21
Journal	Journal d'Analyse Mathématique
Volume	148
Issue number	1
Early online date	17 Nov 2022
DOIs	https://doi.org/10.1007/s11854-022-0231-6
Publication status	Published - 17 Nov 2022

Keywords

Pattern
Density
Similarity
Arithmetic progression
Discrepancy

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10.1007/s11854-022-0231-6

2007.03493Submitted manuscript, 250 KB
Falconer-2021-Density-of-sets-containing-JdAM-AAM
Copyright © 2022, The Hebrew University of Jerusalem. This work has been made available online in accordance with publisher policies or with permission. Permission for further reuse of this content should be sought from the publisher or the rights holder. This is the author created accepted manuscript following peer review and may differ slightly from the final published version. The final published version of this work is available at https://doi.org/10.1007/s11854-022-0231-6.
Accepted author manuscript, 260 KB

https://arxiv.org/abs/2007.03493

Cite this

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title = "The density of sets containing large similar copies of finite sets",

abstract = "We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than (n−2) (n−1), then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1−O(n−1/5log n). ",

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note = "Funding: VK is supported by the Croatian Science Foundation, project n◦ UIP-2017-05-4129 (MUNHANAP). AY is supported by the Swiss National Science Foundation, grant n◦ P2SKP2 184047.",

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AU - Kovač, Vjekoslav

AU - Yavicoli, Alexia

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N2 - We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than (n−2) (n−1), then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1−O(n−1/5log n).

AB - We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than (n−2) (n−1), then E contains similar copies of every n-point set P at all sufficiently large scales. Moreover, 'sufficiently large' can be taken to be uniform over all P with prescribed size, minimum separation and diameter. On the other hand, we construct an example to show that the density required to guarantee all large similar copies of n-point sets tends to 1 at a rate 1−O(n−1/5log n).

KW - Pattern

KW - Density

KW - Similarity

KW - Arithmetic progression

KW - Discrepancy

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EP - 359

JO - Journal d'Analyse Mathématique

JF - Journal d'Analyse Mathématique

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The density of sets containing large similar copies of finite sets

Abstract

Keywords

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