Some toy Furstenberg sets and projections of the four-corner Cantor set

Oberlin, Daniel

doi:10.1090/S0002-9939-2013-11857-5

Some toy Furstenberg sets and projections of the four-corner Cantor set
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by Daniel M. Oberlin

Proc. Amer. Math. Soc. 142 (2014), 1209-1215

DOI: https://doi.org/10.1090/S0002-9939-2013-11857-5

Published electronically: December 26, 2013

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Abstract:

We give lower bounds for the Hausdorff dimensions of some model Furstenberg sets.

References

Michael Bateman and Alexander Volberg, An estimate from below for the Buffon needle probability of the four-corner Cantor set, Math. Res. Lett. 17 (2010), no. 5, 959–967. MR 2727621, DOI 10.4310/MRL.2010.v17.n5.a12
M. Bond, I. Laba, and A. Volberg, Buffon’s needle estimates for rational product Cantor sets, arXiv:1109.1031.
J. Bourgain, On the Erdős-Volkmann and Katz-Tao ring conjectures, Geom. Funct. Anal. 13 (2003), no. 2, 334–365. MR 1982147, DOI 10.1007/s000390300008
Jean Bourgain, The discretized sum-product and projection theorems, J. Anal. Math. 112 (2010), 193–236. MR 2763000, DOI 10.1007/s11854-010-0028-x
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
J.-P. Kahane, Trois notes sur les ensembles parfaits linéaires, Enseign. Math. (2) 15 (1969), 185–192 (French). MR 245734
Robert Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153–155. MR 248779, DOI 10.1112/S0025579300002503
Nets Hawk Katz and Terence Tao, Some connections between Falconer’s distance set conjecture and sets of Furstenburg type, New York J. Math. 7 (2001), 149–187. MR 1856956
F. Nazarov, Y. Peres, and A. Volberg, The power law for the Buffon needle probability of the four-corner Cantor set, Algebra i Analiz 22 (2010), no. 1, 82–97; English transl., St. Petersburg Math. J. 22 (2011), no. 1, 61–72. MR 2641082, DOI 10.1090/S1061-0022-2010-01133-6
Daniel M. Oberlin, Restricted Radon transforms and projections of planar sets, Canad. Math. Bull. 55 (2012), no. 4, 815–820. MR 2994685, DOI 10.4153/CMB-2011-064-6
Elias M. Stein and Rami Shakarchi, Real analysis, Princeton Lectures in Analysis, vol. 3, Princeton University Press, Princeton, NJ, 2005. Measure theory, integration, and Hilbert spaces. MR 2129625
Thomas Wolff, Recent work connected with the Kakeya problem, Prospects in mathematics (Princeton, NJ, 1996) Amer. Math. Soc., Providence, RI, 1999, pp. 129–162. MR 1660476