Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

Request Permissions   Purchase Content 
 
 

 

Are lines much bigger than line segments?


Author: Tamás Keleti
Journal: Proc. Amer. Math. Soc. 144 (2016), 1535-1541
MSC (2010): Primary 28A78
DOI: https://doi.org/10.1090/proc/12978
Published electronically: December 15, 2015
MathSciNet review: 3451230
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We pose the following conjecture:

($ \star $) If $ A$ is the union of line segments in $ \mathbb{R}^n$, and $ B$ is the union of the corresponding full lines, then the Hausdorff dimensions of $ A$ and $ B$ agree.

We prove that this conjecture would imply that every Besicovitch set (compact set that contains line segments in every direction) in $ \mathbb{R}^n$ has Hausdorff dimension at least $ n-1$ and (upper) Minkowski dimension $ n$. We also prove that conjecture ($ \star $) holds if the Hausdorff dimension of $ B$ is at most $ 2$, so in particular it holds in the plane.


References [Enhancements On Off] (What's this?)

  • [1] A. Besicovitch,
    Sur deux questions d'integrabilite des fonctions,
    J. Soc. Phys. Math. 2 (1919), 105-123.
  • [2] Roy O. Davies, Some remarks on the Kakeya problem, Proc. Cambridge Philos. Soc. 69 (1971), 417-421. MR 0272988 (42 #7869)
  • [3] Zeev Dvir, On the size of Kakeya sets in finite fields, J. Amer. Math. Soc. 22 (2009), no. 4, 1093-1097. MR 2525780 (2011a:52039), https://doi.org/10.1090/S0894-0347-08-00607-3
  • [4] M. Elekes,
    private communication.
  • [5] K. J. Falconer, The geometry of fractal sets, Cambridge Tracts in Mathematics, vol. 85, Cambridge University Press, Cambridge, 1986. MR 867284 (88d:28001)
  • [6] Kenneth Falconer, Fractal geometry, 3rd ed., John Wiley & Sons, Ltd., Chichester, 2014. Mathematical foundations and applications. MR 3236784
  • [7] Jonathan M. Fraser, Inhomogeneous self-similar sets and box dimensions, Studia Math. 213 (2012), no. 2, 133-156. MR 3024316, https://doi.org/10.4064/sm213-2-2
  • [8] D. H. Fremlin,
    Measure Theory: Topological Measure Spaces (Vol. 4),
    Torres Fremlin, 2003.
  • [9] Nets Hawk Katz and Terence Tao, New bounds for Kakeya problems, J. Anal. Math. 87 (2002), 231-263. Dedicated to the memory of Thomas H. Wolff. MR 1945284 (2003i:28006), https://doi.org/10.1007/BF02868476
  • [10] Nets Hawk Katz, Izabella Łaba, and Terence Tao, An improved bound on the Minkowski dimension of Besicovitch sets in $ {\bf R}^3$, Ann. of Math. (2) 152 (2000), no. 2, 383-446. MR 1804528 (2002i:28006), https://doi.org/10.2307/2661389
  • [11] I. Łaba and T. Tao, An improved bound for the Minkowski dimension of Besicovitch sets in medium dimension, Geom. Funct. Anal. 11 (2001), no. 4, 773-806. MR 1866801 (2003b:28006), https://doi.org/10.1007/PL00001685
  • [12] D. G. Larman, A compact set of disjoint line segments in $ E^{3}$ whose end set has positive measure, Mathematika 18 (1971), 112-125. MR 0297954 (45 #7006)
  • [13] J. M. Marstrand, Some fundamental geometrical properties of plane sets of fractional dimensions, Proc. London Math. Soc. (3) 4 (1954), 257-302. MR 0063439 (16,121g)
  • [14] Pertti Mattila, Hausdorff dimension, orthogonal projections and intersections with planes, Ann. Acad. Sci. Fenn. Ser. A I Math. 1 (1975), no. 2, 227-244. MR 0409774 (53 #13526)
  • [15] 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 (96h:28006)
  • [16] O. Nikodym,
    Sur le mesure des ensembles plans dont tous les points sont rectalineairément accessibles,
    Fund. Math. 10 (1927), 116-168.
  • [17] Terence Tao, From rotating needles to stability of waves: emerging connections between combinatorics, analysis, and PDE, Notices Amer. Math. Soc. 48 (2001), no. 3, 294-303. MR 1820041 (2002b:42021)
  • [18] Thomas Wolff, An improved bound for Kakeya type maximal functions, Rev. Mat. Iberoamericana 11 (1995), no. 3, 651-674. MR 1363209 (96m:42034), https://doi.org/10.4171/RMI/188
  • [19] 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 (2000d:42010)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 28A78

Retrieve articles in all journals with MSC (2010): 28A78


Additional Information

Tamás Keleti
Affiliation: Institute of Mathematics, Eötvös Loránd University, Pázmány Péter sétány 1/c, H-1117 Budapest, Hungary
Email: tamas.keleti@gmail.com

DOI: https://doi.org/10.1090/proc/12978
Keywords: Hausdorff dimension, lines, union of line segments, Besicovitch set, Nikodym set, Kakeya Conjecture.
Received by editor(s): February 2, 2015
Published electronically: December 15, 2015
Additional Notes: The author was supported by Hungarian Scientific Foundation grant no. 104178. Part of this research was done while the author was a visiting researcher at the Alfréd Rényi Institute of Mathematics, whose hospitality is gratefully acknowledged.
Communicated by: Alexander Iosevich
Article copyright: © Copyright 2015 American Mathematical Society

American Mathematical Society