Hostname: page-component-586b7cd67f-t8hqh Total loading time: 0 Render date: 2024-11-30T04:13:35.701Z Has data issue: false hasContentIssue false
  • English
  • Français

On Hausdorff dimension of projections

Published online by Cambridge University Press:  26 February 2010

Robert Kaufman
Robert Kaufman
Affiliation:
University of Illinois, Urbana, Illinois.
Get access

Extract

Let E be a compact subset of R2, of Hausdorff dimension s > 0 and for each real number t let Ft be the linear set {x1 + tx2: (x1x2) ∈ E}. In this note we shall prove

THEOREM. If s ≤ 1 then Ft has dimension ≥ s, excepting numbers t in a set of dimension ≤ s. If s > 1 then Ft, has positive Lebesgue measure, excepting numbers t in a set of Lebesgue measure 0.

Type
Research Article
Information
Mathematika , Volume 15 , Issue 2 , December 1968 , pp. 153 - 155
Copyright
Copyright © University College London 1968

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1.Davies, R. O., “Subsets of finite measure in analytic sets”, Indag. Math., 14 (1952), 488–489.CrossRefGoogle Scholar
2.Frostman, O., Potientiel d'équilibre et capacityé … (Lund, 1935).Google Scholar
3.Kahane, J.-P. and Salem, R., Ensembles parfaits et séries trigonométriques (Paris, 1963).Google Scholar
4.Marstrand, J. M., “The dimension of Cartesian product sets”, Proc. Cambridge Phil. Soc., 50 (1954), 198–202.CrossRefGoogle Scholar
5.Marstrand, J. M., “Some fundamental geometrical properties of plane sets of fractional dimensions”, Proc. London Math. Soc., 4 (1954), 257–302.CrossRefGoogle Scholar
6.Sion, M. and Sjerve, D., “Approximation properties of measures generated by continuous set functions”, Mathematika, 9 (1962), 145–156.CrossRefGoogle Scholar