Hostname: page-component-cd9895bd7-q99xh Total loading time: 0 Render date: 2024-12-26T16:08:43.861Z Has data issue: false hasContentIssue false
  • English
  • Français

Some dimensional properties of generalised difference sets

Published online by Cambridge University Press:  26 February 2010

D. J. Ward
D. J. Ward
Affiliation:
School of Mathematical and Physical Sciences, The University of Sussex.
Get access

Extract

It is well known that Cantor's ternary set C, constructed on [0, 1], has a difference set, D(C), equal to [0, 1]. If E ⊂ R we define Dk(E)(⊂ Rk-1), by Dk(E) = {(d1, d2, …, dk-1); di ≥ 0, and there is x ∈ E such that x + di ∈ E for all i, 1 ≤ i < k}. Thus D2(E) ≡ D(E) and Dk(E) tells us whether or not a particular set of k real numbers can be translated into E. We call Dk(E) the k-difference set of E. In this work we seek criteria for finding the Besicovitch dimension of Dk(E) (written dim Dk(E))) and in particular, conditions on certain classes of linear sets E that ensure that Dk(E) should contain an open interval in Rk-1.

Type
Research Article
Information
Mathematika , Volume 17 , Issue 2 , December 1970 , pp. 185 - 188
Copyright
Copyright © University College London 1970

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.Besicovitch, A. S., “On linear sets of points of fractional dimension”, Math. Ann., 101 (1929), 161–193.CrossRefGoogle Scholar
2.Besicovitch, A. S., and Moran, P. A. P., “The measure of product and cylinder sets”, J. London Math. Soc., 20 (1945), 110–120.Google Scholar
3.Davies, Roy O., Marstrand, J. M. and Taylor, S. J., “On the intersection of transforms of linear sets”, Coll. Math., 7 (1960), 237–243.CrossRefGoogle Scholar
4.Ward, D. J., “A set of plane measure zero containing all finite polygonal arcs”, Canadian J. Math., 22 (1970), 815–821.CrossRefGoogle Scholar