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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.
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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

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References

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