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Better Bounds for Planar Sets Avoiding Unit Distances

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Abstract

A 1-avoiding set is a subset of \(\mathbb {R}^n\) that does not contain pairs of points at distance 1. Let \(m_1(\mathbb {R}^n)\) denote the maximum fraction of \(\mathbb {R}^n\) that can be covered by a measurable 1-avoiding set. We prove two results. First, we show that any 1-avoiding set in \(\mathbb {R}^n (n\ge 2)\) that displays block structure (i.e., is made up of blocks such that the distance between any two points from the same block is less than 1 and points from distinct blocks lie farther than 1 unit of distance apart from each other) has density strictly less than \(1/2^n\). For the special case of sets with block structure this proves a conjecture of Erdős asserting that \(m_1(\mathbb {R}^2) < 1/4\). Second, we use linear programming and harmonic analysis to show that \(m_1(\mathbb {R}^2) \le 0.258795\).

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Notes

  1. This is related to the following observation: Let G be a subgraph of a finite vertex-transitive graph H. Then \(\alpha (H) / |V(H)| \le \alpha (G) / |V(G)|\).

References

  1. Bachoc, C., Passuello, A., Thiery, A.: The density of sets avoiding distance 1 in Euclidean space. Discrete Comput. Geom. 53(4), 783–808 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  2. Croft, H.T.: Incidence incidents. Eureka 30, 22–26 (1967)

    Google Scholar 

  3. Falconer, K.J.: The realization of distances in measurable subsets covering \({\mathbb{R}}^n\). J. Comb. Theory Ser. A 31, 184–189 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  4. Frankl, P., Wilson, R.M.: Intersection theorems with geometric consequences. Combinatorica 1, 357–368 (1981)

    Article  MathSciNet  MATH  Google Scholar 

  5. Gardner, R.J.: The Brunn–Minkowski inequality. Bull. Am. Math. Soc. 39(3), 355–405 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. Katznelson, Y.: An Introduction to Harmonic Analysis. Wiley, New York (1968)

    MATH  Google Scholar 

  7. Larman, D.G., Rogers, C.A.: The realization of distances within sets in Euclidean space. Mathematika 19, 1–24 (1972)

    Article  MathSciNet  MATH  Google Scholar 

  8. Maggi, F., Ponsiglione, M., Pratelli, A.: Quantitative stability in the isodiametric inequality via the isoperimetric inequality. Trans. Am. Math. Soc. 366, 1141–1160 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  9. Moser, L., Moser, W.: Solution to problem 10. Can. Math. Bull. 4, 187–189 (1961)

    MATH  Google Scholar 

  10. de Oliveira Filho, F.M., Vallentin, F.: Fourier analysis, linear programming, and densities of distance-avoiding sets in \({\mathbb{R}}^n\). J. Eur. Math. Soc. 12, 1417–1428 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  11. Schoenberg, I.J.: Metric spaces and completely monotone functions. Ann. Math. 39, 811–841 (1938)

    Article  MathSciNet  MATH  Google Scholar 

  12. Soifer, A.: The Mathematical Coloring Book. Springer, New York (2009)

    MATH  Google Scholar 

  13. Stein, W.A. et al.: Sage Mathematics Software (Version 6.3), The Sage Development Team (2014). http://www.sagemath.org

  14. Székely, L.A.: Measurable chromatic number of geometric graphs and sets without some distances in Euclidean space. Combinatorica 4, 213–218 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  15. Székely, L.A.: Erdős on unit distances and the Szemerédi–Trotter theorems. In: Halász, G., Lovász, L., Simonovits, M., Sós, V.T. (eds.) Paul Erdős and His Mathematics II. Bolyai Society Mathematical Studies 11. János Bolyai Mathematical Society, Budapest, pp. 646–666. Springer, Berlin (2002)

  16. Székely, L.A., Wormald, N.C.: Bounds on the measurable chromatic number of \(R^n\). Discrete Math. 75, 343–372 (1989)

    Article  MathSciNet  MATH  Google Scholar 

  17. Watson, G.N.: A Treatise on the Theory of Bessel Functions. Cambridge University Press, Cambridge (1922)

    MATH  Google Scholar 

Download references

Acknowledgments

The authors are grateful to the referees whose valuable suggestions helped to improve the presentation of the paper. Part of this research was done when T. Keleti was a visitor at the Alfréd Rényi Institute of Mathematics; he was also supported by OTKA Grant No. 104178. M. Matolcsi and I.Z. Ruzsa were supported by OTKA No. 109789 and ERC-AdG 321104. F.M. de Oliveira Filho was partially supported by FAPESP project 13/03447-6.

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Correspondence to Máté Matolcsi.

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Keleti, T., Matolcsi, M., de Oliveira Filho, F.M. et al. Better Bounds for Planar Sets Avoiding Unit Distances. Discrete Comput Geom 55, 642–661 (2016). https://doi.org/10.1007/s00454-015-9751-5

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  • DOI: https://doi.org/10.1007/s00454-015-9751-5

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