Quantitative visibility estimates for unrectifiable sets in the plane

Bond, M.; Łaba, I.; Zahl, J.

doi:10.1090/tran/6585

Quantitative visibility estimates for unrectifiable sets in the plane
HTML articles powered by AMS MathViewer

by M. Bond, I. Łaba and J. Zahl PDF

Trans. Amer. Math. Soc. 368 (2016), 5475-5513 Request permission

Abstract:

The “visibility” of a planar set $S$ from a point $a$ is defined as the normalized size of the radial projection of $S$ from $a$ to the unit circle centered at $a$. Simon and Solomyak in 2006 proved that unrectifiable self-similar one-sets are invisible from every point in the plane. We quantify this by giving an upper bound on the visibility of $\delta$-neighborhoods of such sets. We also prove lower bounds on the visibility of $\delta$-neighborhoods of more general sets, based in part on Bourgain’s discretized sum-product estimates.

References

Ida Arhosalo, Esa Järvenpää, Maarit Järvenpää, MichałRams, and Pablo Shmerkin, Visible parts of fractal percolation, Proc. Edinb. Math. Soc. (2) 55 (2012), no. 2, 311–331. MR 2928497, DOI 10.1017/S0013091509001680
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
A. S. Besicovitch, On the fundamental geometrical properties of linearly measurable plane sets of points (III), Math. Ann. 116 (1939), no. 1, 349–357. MR 1513231, DOI 10.1007/BF01597361
Matthew Robert Bond, Combinatorial and Fourier analytic L2 methods for Buffon’s needle problem, ProQuest LLC, Ann Arbor, MI, 2011. Thesis (Ph.D.)–Michigan State University. MR 2941771
Matthew Bond, Izabella Łaba, and Alexander Volberg, Buffon’s needle estimates for rational product Cantor sets, Amer. J. Math. 136 (2014), no. 2, 357–391. MR 3188064, DOI 10.1353/ajm.2014.0013
Matthew Bond and Alexander Volberg, Buffon needle lands in $\epsilon$-neighborhood of a 1-dimensional Sierpinski gasket with probability at most $|\log \epsilon |^{-c}$, C. R. Math. Acad. Sci. Paris 348 (2010), no. 11-12, 653–656 (English, with English and French summaries). MR 2652491, DOI 10.1016/j.crma.2010.04.006
Matthew Bond and Alexander Volberg, Buffon’s needle landing near Besicovitch irregular self-similar sets, Indiana Univ. Math. J. 61 (2012), no. 6, 2085–2109. MR 3129103, DOI 10.1512/iumj.2012.61.4828
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
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
Marianna Csörnyei, On the visibility of invisible sets, Ann. Acad. Sci. Fenn. Math. 25 (2000), no. 2, 417–421. MR 1762426
Marianna Csörnyei, How to make Davies’ theorem visible, Bull. London Math. Soc. 33 (2001), no. 1, 59–66. MR 1798576, DOI 10.1112/blms/33.1.59
P. Erdős and P. Révész, On the length of the longest head-run, Topics in information theory (Second Colloq., Keszthely, 1975) Colloq. Math. Soc. János Bolyai, Vol. 16, North-Holland, Amsterdam, 1977, pp. 219–228. MR 0478304
Kemal Ilgar Eroğlu, On planar self-similar sets with a dense set of rotations, Ann. Acad. Sci. Fenn. Math. 32 (2007), no. 2, 409–424. MR 2337485
K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284
Kenneth J. Falconer and Jonathan M. Fraser, The visible part of plane self-similar sets, Proc. Amer. Math. Soc. 141 (2013), no. 1, 269–278. MR 2988729, DOI 10.1090/S0002-9939-2012-11312-7
John Garnett, Positive length but zero analytic capacity, Proc. Amer. Math. Soc. 24 (1970), 696-699; errata, ibid. 26 (1970), 701. MR 0276456, DOI 10.1090/S0002-9939-1970-0276456-5
M. Hochman, Dynamics on fractals and fractal distributions, arXiv: http://xxx.lanl.gov/abs/1008.3731
Michael Hochman, On self-similar sets with overlaps and inverse theorems for entropy, Ann. of Math. (2) 180 (2014), no. 2, 773–822. MR 3224722, DOI 10.4007/annals.2014.180.2.7
Michael Hochman and Pablo Shmerkin, Local entropy averages and projections of fractal measures, Ann. of Math. (2) 175 (2012), no. 3, 1001–1059. MR 2912701, DOI 10.4007/annals.2012.175.3.1
M. Hochman and P. Shmerkin, Equidistribution from fractal measures, arXiv: http://xxx.lanl.gov/abs/1302.5792
John E. Hutchinson, Fractals and self-similarity, Indiana Univ. Math. J. 30 (1981), no. 5, 713–747. MR 625600, DOI 10.1512/iumj.1981.30.30055
Esa Järvenpää, Maarit Järvenpää, Paul MacManus, and Toby C. O’Neil, Visible parts and dimensions, Nonlinearity 16 (2003), no. 3, 803–818. MR 1975783, DOI 10.1088/0951-7715/16/3/302
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
Robert Kaufman, On Hausdorff dimension of projections, Mathematika 15 (1968), 153–155. MR 248779, DOI 10.1112/S0025579300002503
Richard Kenyon, Projecting the one-dimensional Sierpinski gasket, Israel J. Math. 97 (1997), 221–238. MR 1441250, DOI 10.1007/BF02774038
Izabella Łaba and Kelan Zhai, The Favard length of product Cantor sets, Bull. Lond. Math. Soc. 42 (2010), no. 6, 997–1009. MR 2740020, DOI 10.1112/blms/bdq059
Jeffrey C. Lagarias and Yang Wang, Tiling the line with translates of one tile, Invent. Math. 124 (1996), no. 1-3, 341–365. MR 1369421, DOI 10.1007/s002220050056
Pertti Mattila, Integral geometric properties of capacities, Trans. Amer. Math. Soc. 266 (1981), no. 2, 539–554. MR 617550, DOI 10.1090/S0002-9947-1981-0617550-8
Pertti Mattila, Orthogonal projections, Riesz capacities, and Minkowski content, Indiana Univ. Math. J. 39 (1990), no. 1, 185–198. MR 1052016, DOI 10.1512/iumj.1990.39.39011
Pertti Mattila, Geometry of sets and measures in Euclidean spaces, Cambridge Studies in Advanced Mathematics, vol. 44, Cambridge University Press, Cambridge, 1995. Fractals and rectifiability. MR 1333890, DOI 10.1017/CBO9780511623813
Pertti Mattila, Hausdorff dimension, projections, and the Fourier transform, Publ. Mat. 48 (2004), no. 1, 3–48. MR 2044636, DOI 10.5565/PUBLMAT_{4}8104_{0}1
J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3) 4 (1954), 257–302. MR 63439, DOI 10.1112/plms/s3-4.1.257
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
Toby C. O’Neil, The Hausdorff dimension of visible sets of planar continua, Trans. Amer. Math. Soc. 359 (2007), no. 11, 5141–5170. MR 2327025, DOI 10.1090/S0002-9947-07-04460-1
Yuval Peres and Boris Solomyak, How likely is Buffon’s needle to fall near a planar Cantor set?, Pacific J. Math. 204 (2002), no. 2, 473–496. MR 1907902, DOI 10.2140/pjm.2002.204.473
Károly Simon and Boris Solomyak, Visibility for self-similar sets of dimension one in the plane, Real Anal. Exchange 32 (2006/07), no. 1, 67–78. MR 2329222, DOI 10.14321/realanalexch.32.1.0067
Xavier Tolsa, Analytic capacity, rectifiability, and the Cauchy integral, International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 1505–1527. MR 2275656
Thomas H. Wolff, Lectures on harmonic analysis, University Lecture Series, vol. 29, American Mathematical Society, Providence, RI, 2003. With a foreword by Charles Fefferman and a preface by Izabella Łaba; Edited by Łaba and Carol Shubin. MR 2003254, DOI 10.1090/ulect/029