Skip to main content
×
Home

On the Hausdorff dimensions of distance sets

  • K. J. Falconer (a1)
Abstract

If E is a subset of ℝn (n ≥ 1) we define the distance set of E as

The best known result on distance sets is due to Steinhaus [11], namely, that, if E ⊂ ℝn is measurable with positive n-dimensional Lebesgue measure, then D(E) contains an interval [0, ε) for some ε > 0. A number of variations of this have been examined, see Falconer [6, p. 108] and the references cited therein.

Copyright
References
Hide All
1.Carleson L.. Selected Problems on Exceptional Sets (Van Nostrand, Princeton, 1967).
2.Chung F. R. K.. The number of different distances determined by n points in the plane J. Combinatorial Theory A, 26 (1984), 342354.
3.Davies Roy O.. Subsets of finite measure in analytic sets. Indag. Math., 14 (1952), 488489.
4.Davies Roy O.. Two counter-examples concerning Hausdorff dimensions of projections. Colloq. Math., 42 (1979), 5358.
5.Davies Roy O.. Rings of dimension d. To appear.
6.Falconer K. J.. The Geometry of Fractal Sets (Cambridge University Press, 1984).
7.Falconer K. J.. Rings of fractional dimension. Mathematika, 31 (1984), 2527.
8.Falconer K. J.. Classes of sets with large intersection. Mathematika, 32 (1985), 191205.
9.Marstrand J. M.. The dimension of cartesian product sets. Proc. Cambridge Phil. Soc., 50 (1954), 198202.
10.Rogers C. A.. Hausdorff Measures (Cambridge University Press, 1970).
11.Steinhaus H.. Sur les distances des points des ensembles de mesure positive. Fund. Math., 1 (1920), 93104.
12.Watson G. N.. A Treatise on the Theory of Bessel Functions (Cambridge University Press, 1984).
Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *
×
MathJax

Keywords:

Metrics

Full text views

Total number of HTML views: 0
Total number of PDF views: 70 *
Loading metrics...

Abstract views

Total abstract views: 287 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 21st November 2017. This data will be updated every 24 hours.