The density of sets containing large similar copies of finite sets
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We prove that if E⊆Rd (d≥2) is a Lebesgue-measurable set with density larger than n−2n−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).
Falconer , K , Yavicoli , A & Kovac , V 2021 , ' The density of sets containing large similar copies of finite sets ' , Journal d'Analyse Mathématique . < https://arxiv.org/abs/2007.03493 >
Journal d'Analyse Mathématique
DescriptionFunding: 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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