# On sets of directions determined by subsets of ℝ^{d}

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## Abstract

Given to be the set of directions determined by

*E*⊂ ℝ^{d},*d*≥ 2, define$$D(E) \equiv \left\{ {{{x - y} \over {\left| {x - y} \right|}}:x,y \in E} \right\} \subset {S^{d - 1}}$$

*E*. We prove that if the Hausdorff dimension of*E*is greater than*d*− 1, then*σ*(*D*(*E*)) > 0, where*σ*denotes the surface measure on*S*^{d−1}. In the process, we prove some tight upper and lower bounds for the maximal function associated with the Radon-Nikodym derivative of the natural measure on*D*. This result is sharp, since the conclusion fails to hold if*E*is a (*d*− 1)-dimensional hyper-plane. This result can be viewed as a continuous analog of a recent result of Pach, Pinchasi, and Sharir ([22, 23]) on directions determined by finite subsets of ℝ^{d}. We also discuss the case when the Hausdorff dimension of*E*is precisely*d*− 1, where some interesting counter-examples have been obtained by Simon and Solomyak ([25]) in the planar case. In response to the conjecture stated in this paper, T. Orponen and T. Sahlsten ([20]) have recently proved that if the Hausdorff dimension of*E*equals*d*− 1 and*E*is rectifiable and is not contained in a hyper-pane, the Lebesgue measure of the set of directions is still positive. Finally, we show that our continuous results can be used to recover and, in some cases, improve the exponents for the corresponding results in the discrete setting for large classes of finite point sets. In particular, we prove that a finite point set*P*⊂ ℝ^{d},*d*≥ 3, satisfying a certain discrete energy condition (Definition 3.1) determines ≳ #*P*distinct directions.## Preview

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### References

- [1]P. Brass, W. Moser, and J. Pach,
*Research Problems in Discrete Geometry*, Springer, New York, 2005.MATHGoogle Scholar - [2]J. Bourgain,
*A Szemerédi type theorem for sets of positive density*, Israel J. Math.**54**(1986), 307–331.MathSciNetMATHCrossRefGoogle Scholar - [3]J. Bourgain,
*Hausdorff dimension and distance sets*, Israel. J. Math.**87**(1994), 193–201.MathSciNetMATHCrossRefGoogle Scholar - [4]D. Covert, B. Erdogan, D. Hart, A. Iosevich, and K. Taylor
*Finite point configurations, uniform distribution, intersections of fractals, and number theoretic consequences*, in preparation.Google Scholar - [5]M. B. Erdoğan
*A bilinear Fourier extension theorem and applications to the distance set problem*, Int. Math. Res. Not.**2005**, 1411–1425.Google Scholar - [6]K. J. Falconer
*On the Hausdorff dimensions of distance sets*, Mathematika**32**(1986), 206–212.MathSciNetCrossRefGoogle Scholar - [7]K. J. Falconer,
*The Geometry of Fractal Sets*, Cambridge University Press, Cambridge, 1986.Google Scholar - [8]H. Furstenberg, Y. Katznelson, and B. Weiss,
*Ergodic theory and configurations in sets of positive density*Mathematics of Ramsey Theory, Springer, Berlin, 1990, pp. 184–198.Google Scholar - [9]J. Garnett and J. Verdera,
*Analytic capacity, bilipschitz maps and Cantor sets*, Math. Res. Lett.**10**(2003), 515–522.MathSciNetMATHGoogle Scholar - [10]D. Hart, A. Iosevich, D. Koh, and M. Rudnev,
*Averages over hyperplanes, sum-product theory in vector spaces over finite fields and the Erdős-Falconer distance conjecture*, Trans. Amer. Math. Soc.**363**(2011), 3255–3275.MathSciNetMATHCrossRefGoogle Scholar - [11]S. Hofmann and A. Iosevich,
*Circular averages and Falconer/Erdős distance conjecture in the plane for random metrics*, Proc. Amer. Math. Soc.**133**(2005), 133–143.MathSciNetMATHCrossRefGoogle Scholar - [12]A. Iosevich and I. Laba,
*K-distance sets, Falconer conjecture, and discrete analogs*, Integers**5**(2005), A8.MathSciNetGoogle Scholar - [13]A. Iosevich, H. Morgan and J. Pakianathan,
*Sets of directions determined by subsets of vector spaces over finite fields*, Integers**11**(2011), A39.CrossRefGoogle Scholar - [14]A. Iosevich, M. Rudnev, and I. Uriarte-Tuero,
*Theory of dimension for large discrete sets and applications*, arXiv:0707.1322.Google Scholar - [15]A. Iosevich and S. Senger,
*Sharpness of Falconer’s estimate in continuous and arithmetic settings, geometric incidence theorems and distribution of lattice points in convex domains*, preprint, 2010.Google Scholar - [16]E. Landau,
*Vorlesungenüber Zahlentheorie*, Chelsea Publishing Company, New York, 1969.Google Scholar - [17]J. Matoušek,
*Lectures on Discrete Geometry*, Springer, New York, 2002.MATHGoogle Scholar - [18]P. Mattila,
*Spherical averages of Fourier transforms of measures with finite energy: dimensions of intersections and distance sets*, Mathematika**34**(1987), 207–228.MathSciNetMATHCrossRefGoogle Scholar - [19]P. Mattila,
*Geometry of Sets and Measures in Euclidean Spaces*, Cambridge University Press, Cambridge, 1995.MATHGoogle Scholar - [20]T. Orponen and T. Sahlsten,
*Radial projections of rectifiable sets*, Ann. Acad. Sci. Fenn. Math.**36**(2011), 677–681.MathSciNetMATHCrossRefGoogle Scholar - [21]J. Pach,
*Directions in combinatorial geometry*, Jahresber. Deutsch. Math.-Verein.**107**(2005), 215–225.MathSciNetMATHGoogle Scholar - [22]J. Pach, R. Pinchasi, and M. Sharir,
*On the number of directions determined by a three-dimensional points set*, J. Combin. Theory Ser. A**108**(2004), 1–16.MathSciNetMATHCrossRefGoogle Scholar - [23]J. Pach, R. Pinchasi, and M. Sharir,
*Solution of Scott’s problem on the number of directions determined by a point set in 3-space*, Discrete Comput. Geom.**38**(2007), 399–441.MathSciNetMATHCrossRefGoogle Scholar - [24]J. Pach and M. Sharir,
*Geometric incidences*, Towards a Theory of Geometric Graphs, Amer. Math. Soc., Providence, RI, 2004, pp. 185–223.Google Scholar - [25]K. Simon and B. Solomyak,
*Visibility for self-similar sets of dimension one in the plane*, Real Anal. Exchange**32**(2006/07), 67–78.MathSciNetGoogle Scholar - [26]L. Székely,
*Crossing numbers and hard Erdős problems in discrete geometry*, Combin. Probab. Comput.**6**(1997), 353–358.MathSciNetMATHCrossRefGoogle Scholar - [27]T. Wolff,
*Decay of circular means of Fourier transforms of measures*, Internat. Math. Res. Not.**1999**, 547–567.Google Scholar - [28]T. Wolff,
*Recent work connected with the Kakeya problem*, Prospects in Mathematics, Amer. Math. Soc., Providence, RI, 1999, pp. 129–162.Google Scholar - [29]T. Ziegler,
*Nilfactors of*ℝ^{d}*actions and configurations in sets of positive upper density in*ℝ^{m}, J. Anal. Math.**99**(2006), 249–266.MathSciNetMATHCrossRefGoogle Scholar

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