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The density of sets containing large similar copies of finite sets

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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−1/5 log n).

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References

  1. D. Bilyk, Discrepancy theory and harmonic analysis, in Uniform Distribution and Quasi-Monte Carlo Methods, De Gruyter, Berlin, 2014, pp. 45–61.

    Chapter  Google Scholar 

  2. J. Bourgain, A Szemerédi type theorem for sets of positive density in Rk, Israel J. Math. 54 (1986), 307–316.

    Article  MathSciNet  MATH  Google Scholar 

  3. Y. Bugeaud, Distribution Modulo one and Diophantine Approximation, Cambridge University Press, Cambridge, 2012.

    Book  MATH  Google Scholar 

  4. V. Chan, I. Łaba and M. Pramanik, Finite configurations in sparse sets, J. Anal. Math. 128 (2016), 289–335.

    Article  MathSciNet  MATH  Google Scholar 

  5. B. Cook, Á. Magyar and M. Pramanik, A Roth-type theorem for dense subsets of ℝd, Bull. Lond. Math. Soc. 49 (2017), 676–689.

    Article  MathSciNet  MATH  Google Scholar 

  6. M. Drmota and R. F. Tichy, Sequences, Discrepancies and Applications, Springer, Berlin, 1997.

    Book  MATH  Google Scholar 

  7. P. Durcik and V. Kovač, Boxes, extended boxes, and sets of positive upper density in the Euclidean space, Math. Proc. Cambridge Philos. Soc. 171 (2021), 481–501.

    Article  MathSciNet  MATH  Google Scholar 

  8. P. Durcik and V. Kovač, A Szemerédi-type theorem for subsets of the unit cube, Anal. PDE. 15 (2022), 507–549.

    Article  MathSciNet  MATH  Google Scholar 

  9. P. Durcik, V. Kovač and L. Rimanić, On side lengths of corners in positive density subsets of the Euclidean space, Int. Math. Res. Not. IMRN 22 (2018), 6844–6869.

    Article  MathSciNet  MATH  Google Scholar 

  10. K. J. Falconer and J. M. Marstrand, Plane sets with positive density at infinity contain all large distances, Bull. London Math. Soc. 18 (1986), 471–474.

    Article  MathSciNet  MATH  Google Scholar 

  11. H. Furstenberg, Y. Katznelson and B. Weiss, Ergodic theory and configurations in sets of positive density, in Mathematics of Ramsey Theory, Springer, Berlin, 1990, pp. 184–198.

    Chapter  MATH  Google Scholar 

  12. R. L. Graham, Recent trends in Euclidean Ramsey theory, Discrete Math. 136 (1994), 119–127.

    Article  MathSciNet  MATH  Google Scholar 

  13. A. Greenleaf, A. Iosevich, B. Liu and E. Palsson, An elementary approach to simplexes in thin subsets of euclidean space, arXiv:1608.04777 [math.CA].

  14. A. Greenleaf, A. Iosevich and S. Mkrtchyan, Existence of similar point configurations in thin subsets of ℝd, Math. Z. 297 (2021), 855–865.

    Article  MathSciNet  MATH  Google Scholar 

  15. K. Henriot, I. Łaba and M. Pramanik, On polynomial configurations in fractal sets, Anal. PDE 9 (2016), 1153–1184.

    Article  MathSciNet  MATH  Google Scholar 

  16. L. Huckaba, N. Lyall and Á. Magyar, Simplices and sets of positive upper density in ℝd, Proc. Amer. Math. Soc. 145 (2017), 2335–2347.

    Article  MathSciNet  MATH  Google Scholar 

  17. A. Iosevich and B. Liu, Equilateral triangles in subsets of ℝdof large Hausdorff dimension, Israel J. Math. 231 (2019), 123–137.

    Article  MathSciNet  MATH  Google Scholar 

  18. A. Iosevich and K. Taylor, Finite trees inside thin subsets of ℝd, in Modern Methods in Operator Theory and Harmonic Analysis, Springer, Cham, 2019, 51–56.

    Chapter  Google Scholar 

  19. M. Kolountzakis, Distance sets corresponding to convex bodies, Geom. Funct. Anal. 14 (2004), 734–744.

    Article  MathSciNet  MATH  Google Scholar 

  20. V. Kovač, Density theorems for anisotropic point configurations, Canad. J. Math., to appear.

  21. L. Kuipers and H. Niederreiter, Uniform Distribution of Sequences, Wiley-Interscience, New York-London-Sydney, 1974.

    MATH  Google Scholar 

  22. I. Łaba and M. Pramanik, Arithmetic progressions in sets of fractional dimension. Geom. Funct. Anal. 19 (2009), 429–456.

    Article  MathSciNet  MATH  Google Scholar 

  23. N. Lyall and Á. Magyar, Product of simplices and sets of positive upper density in ℝd, Math. Proc. Cambridge Philos. Soc. 165 (2018), 25–51.

    Article  MathSciNet  MATH  Google Scholar 

  24. N. Lyall and Á. Magyar, Distance graphs and sets of positive upper density in ℝd, Anal. PDE 13 (2020), 685–700.

    Article  MathSciNet  MATH  Google Scholar 

  25. N. Lyall and Á. Magyar, Weak hypergraph regularity and applications to geometric Ramsey theory, Trans. Amer. Math. Soc. Ser. B 9 (2022), 160–207

    Article  MathSciNet  MATH  Google Scholar 

  26. I. D. Morris, A note on configurations in sets of positive density which occur at all large scales, Israel J. Math. 207 (2015), 719–738.

    Article  MathSciNet  MATH  Google Scholar 

  27. A. Quas, Distances in positive density sets in ℝd, J. Combin. Theory Ser. A 116 (2009), 979–987.

    Article  MathSciNet  MATH  Google Scholar 

  28. A. Rice, Sets in ℝdwith slow-decaying density that avoid an unbounded collection of distances, Proc. Amer. Math. Soc. 148 (2020), 523–526.

    Article  MathSciNet  MATH  Google Scholar 

  29. L. Székely, Remarks on the chromatic number of geometric graphs, in Graphs and Other Combinatorial Topics (Prague, 1982), Teubner, Leipzig, 1983, pp. 312–315.

    Google Scholar 

  30. T. Tao. Higher Order Fourier Analysis, American Mathematical Society, Providence, RI, 2012.

    Book  MATH  Google Scholar 

  31. A. Yavicoli, Patterns in thick compact sets, Israel J. Math. 244 (2021), 95–126.

    Article  MathSciNet  MATH  Google Scholar 

  32. T. Ziegler, Nilfactors of ℝm-actions and configurations in sets of positive upper density in ℝm, J. Anal. Math. 99 (2006), 249–266.

    Article  MathSciNet  Google Scholar 

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Author notes

  1. Alexia Yavicoli

    Present address: Current address Department of Mathematics, the University of British Columbia, 1984 Mathematics Road, Vancouver, BC, V6T 1Z2, Canada

Authors and Affiliations

  1. School of Mathematics and Statistics, University of St. Andrews, North Haugh, St Andrews, KY16 9SS, Scotland, UK

    Kenneth Falconer & Alexia Yavicoli

  2. Department of Mathematics Faculty of Science, University of Zagreb, Bijenička cesta 30, 10000, Zagreb, Croatia

    Vjekoslav Kovač

Authors
  1. Kenneth Falconer
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  2. Vjekoslav Kovač
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  3. Alexia Yavicoli
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Corresponding author

Correspondence to Alexia Yavicoli.

Additional information

V. Kovač is supported by the Croatian Science Foundation, project n° UTP-2017-05-4129 (MUNHANAP)

A. Yavicoli is supported by the Swiss National Science Foundation, grant n° P2SKP2_184047

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Falconer, K., Kovač, V. & Yavicoli, A. The density of sets containing large similar copies of finite sets. JAMA 148, 339–359 (2022). https://doi.org/10.1007/s11854-022-0231-6

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  • DOI: https://doi.org/10.1007/s11854-022-0231-6

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