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On sets of fractional dimensions (II)
Published online by Cambridge University Press: 24 October 2008
Extract
1. Any number x between 0 and 1 may be expressed uniquely in the form
where xr is a non-negative integer less than r (r = 2,3,…). We consider the set E of numbers x for which
We establish an inequality connecting the dimensional number of the set E with certain constants of the series
in particular we show that, when ξτ = rθ, the dimensional number of E is θ. We are concerned with the measure of Hausdorff.
- Type
- Research Article
- Information
- Mathematical Proceedings of the Cambridge Philosophical Society , Volume 37 , Issue 2 , April 1941 , pp. 127 - 133
- Copyright
- Copyright © Cambridge Philosophical Society 1941
References
† Hausdorff, F., “Dimension und äusseres Mass”, Math. Annalen, 79 (1919), 157–79.CrossRefGoogle Scholar
‡ Besicovitch, A. S., “Linear sets of points of fractional dimensions”, Math. Annalen, 101 (1929), 161–93.CrossRefGoogle Scholar
§ We write s − m*E in place of t s − m*E where there can be no ambiguity.
† Best, , “A closed dimensionless linear set”, Proc. Edinburgh Math. Soc. (2), 6 (1939), 105–8.CrossRefGoogle Scholar
† Gillis, J., “Note on a theorem of Myrberg”, Proc. Cambridge Phil. Soc. 33 (1937), 419–24CrossRefGoogle Scholar
† For if 0 <f−mE<∞ and 
then g−mE = ∞.
† I am indebted to the referee here for a considerable simplification of the argument.