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
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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