The density of sets containing large similar copies of finite sets
Date
17/11/2022Metadata
Show full item recordAbstract
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).
Citation
Falconer , K , Kovač , V & Yavicoli , A 2022 , ' The density of sets containing large similar copies of finite sets ' , Journal d'Analyse Mathématique , vol. 148 , no. 1 , pp. 339-359 . https://doi.org/10.1007/s11854-022-0231-6
Publication
Journal d'Analyse Mathématique
Status
Peer reviewed
ISSN
0021-7670Type
Journal article
Rights
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.
Description
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.Collections
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