Skip to main content
SpringerLink
Log in

Patterns in thick compact sets

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

We introduce a connection between Newhouse thickness and patterns through a variant of Schmidt’s game introduced by Broderick, Fishman and Simmons. This yields an explicit, robust and checkable condition that ensures that a Cantor set in the real line contains long arithmetic progressions and, more generally, homothetic copies of large finite sets.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. S. Astels, Cantor sets and numbers with restricted partial quotients, Transactions of the American Mathematical Society 352 (2000), 133–170.

    Article  MathSciNet  Google Scholar 

  2. R. Broderick, L. Fishman and D. Simmons, Quantitative results using variants of Schmidt’s game: dimension bounds, arithmetic progressions, and more, Acta Arithmetica 188 (2019), 289–316.

    Article  MathSciNet  Google Scholar 

  3. C. A. Cabrelli, K. E. Hare and U. M. Molter, Sums of Cantor sets, Ergodic Theory and Dynamiacl Systems 17 (1997), 1299–1313.

    Article  MathSciNet  Google Scholar 

  4. V. Chan, I. Łaba and M. Pramanik, Finite configurations in sparse sets, Journal d’Analyse Mathématique 128 (2016), 289–335.

    Article  MathSciNet  Google Scholar 

  5. R. O. Davies, J. M. Marstrand and S. J. Taylor, On the intersections of transforms of linear sets, Colloquium Mathematicum 7 (1959/60), 237–243.

    Article  MathSciNet  Google Scholar 

  6. K. Falconer, Fractal Geometry, John Wiley & Sons, Chichester, 2014.

    MATH  Google Scholar 

  7. R. Fraser, S. Guo and M. Pramanik, Polynomial Roth theorems on sets of fractional dimensions, https://arxiv.org/abs/1904.11123.

  8. A. Greenleaf, A. Iosevich, B. Liu and E. Palsson, A group-theoretic viewpoint on Erdős-Falconer problems and the Mattila integral, Revista Matemática Iberoamerican 31 (2015), 799–810.

    Article  Google Scholar 

  9. A. Greenleaf, A. Iosevich, B. Liu and E. Palsson, An elementary approach to simplexes in thin subsets of euclidean space, https://arxiv.org/abs/1608.04777.

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

    Article  MathSciNet  Google Scholar 

  11. L. Guth, A. Iosevich, Y. Ou and H. Wang, On Falconer’s distance set problem in the plane, Inventiones Mathematicae 219 (2020), 779–830.

    Article  MathSciNet  Google Scholar 

  12. V. Harangi, T. Keleti, G. Kiss, P. Maga, A. Máthé, P. Mattila and B. Strenner, How large dimension guarantees a given angle?, Monatshefte für Mathematik 171 (2013), 169–187.

    Article  MathSciNet  Google Scholar 

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

    Article  MathSciNet  Google Scholar 

  14. B. R. Hunt, I. Kan and J. A. Yorke, When Cantor sets intersect thickly, Transactions of the American Mathematical Society 339 (1993), 869–888.

    Article  MathSciNet  Google Scholar 

  15. A. Iosevich and B. Liu, Equilateral triangles in subsets of ℝd of large Hausdorff dimension, Israel Journal of Mathematics 231 (2019), 123–137.

    Article  MathSciNet  Google Scholar 

  16. A. Iosevich and K. Taylor, Finite trees inside thin subsets of ℝd, in Modern Methods in Operator Theory and Harmonic Analysis, Springer Proceedings in Mathematics & Statistics, Vol. 291, Springer, Cham, 2019, pp. 51–56.

    Chapter  Google Scholar 

  17. T. Keleti, A 1-dimensional subset of the reals that intersects each of its translates in at most a single point, Real Analysis Exchange 24 (1998/99), 843–844.

    Article  MathSciNet  Google Scholar 

  18. T. Keleti, Construction of one-dimensional subsets of the reals not containing similar copies of given patterns, Analysis & PDE 1 (2008), 29–33.

    Article  MathSciNet  Google Scholar 

  19. B. Krause, A non-linear Roth theorem for fractals of sufficiently large dimension, https://arxiv.org/abs/1904.10562.

  20. I. Łaba and M. Pramanik, Arithmetic progressions in sets of fractional dimension, Geometric and Functional Analysis 19 (2009), 429–456.

    Article  MathSciNet  Google Scholar 

  21. A. Máthé, Sets of large dimension not containing polynomial configurations, Advances in Mathematics 316 (2017), 691–709.

    Article  MathSciNet  Google Scholar 

  22. U. Molter and A. Yavicoli, Small sets containing any pattern, Mathematical Proceedings of the Cambridge Philosophical Society 168 (2020), 57–73.

    Article  MathSciNet  Google Scholar 

  23. S. E. Newhouse, Nondensity of axiom A(a) on S2, in Global Analysis, Proceedings of Symposia in Pure Mathematics, Vol. 14, American Mathematical Society, Providence, RI, 1970, pp. 191–202.

    Chapter  Google Scholar 

  24. J. Palis and F. Takens, Hyperbolicity and Sensitive Chaotic Dynamics at Homoclinic Bifurcations, Cambridge Studies in Advanced Mathematics, Vol. 35, Cambridge University Press, Cambridge, 1993.

    MATH  Google Scholar 

  25. Y. Peres and P. Shmerkin, Resonance between Cantor sets, Ergodic Theory and Dynamical Systems 29 (2009), 201–221.

    Article  MathSciNet  Google Scholar 

  26. Y. Peres and B. Solomyak, Self-similar measures and intersections of Cantor sets, Transactions of the American Mathematical Society 350 (1998), 4065–4087.

    Article  MathSciNet  Google Scholar 

  27. A. Yavicoli, Large sets avoiding linear patterns, Proceedings of the American Mathematical Society, to appear, https://doi.org/10.1090/proc/13959.

Download references

Author information

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, KY, 16 9SS, UK

    Alexia Yavicoli

Authors
  1. Alexia Yavicoli
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alexia Yavicoli.

Additional information

This work was partially supported by CONICET and the Swiss National Science Foundation, grant n° P2SKP2_184047.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Yavicoli, A. Patterns in thick compact sets. Isr. J. Math. 244, 95–126 (2021). https://doi.org/10.1007/s11856-021-2173-6

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-021-2173-6

Search

Navigation