The density of sets containing large similar copies of finite sets

Kenneth Falconer, Vjekoslav Kovač, Alexia Yavicoli*

*Corresponding author for this work

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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).
Original languageEnglish
Pages (from-to)339-359
Number of pages21
JournalJournal d'Analyse Mathématique
Issue number1
Early online date17 Nov 2022
Publication statusPublished - 17 Nov 2022


  • Pattern
  • Density
  • Similarity
  • Arithmetic progression
  • Discrepancy


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