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Some dimensional properties of generalised difference sets
Published online by Cambridge University Press: 26 February 2010
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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.
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- Copyright © University College London 1970
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