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A very sparse set of dimension 1

Part of: Classical measure theory

Published online by Cambridge University Press:  26 February 2010

Anatole Beck
Anatole Beck
Affiliation:
Mathematics Department, University of Wisconsin-Madison, Madison, Wisconsin, 53706, U.S.A.
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Extract

In July 1982, I was asked by Prof. Jorgen Hoffmann-Jorgensen to construct an uncountable compact set K in the line which was symmetric about 0 and had the property that, for all n, the set of sums of n-tuples from K has measure 0. There are two equivalent conditions: the set of such sums should never contain an interval, or K* ≠ ℝ, where K* is the subgroup of (ℝ, +) generated by K. I did so, and the set I constructed had entropy dimension 0 (and thus also Hausdorff dimension 0). Hoffmann-Jorgensen showed that every set of entropy dimension 0 would exhibit the same behaviour. However, I did not believe that the essence of the example lay in its dimension, and I here modify my construction so that the set K has dimension 1 (and thus also entropy dimension 1), while K* ≠ ℝ, as before. By contrast, the Cantor ternary set has dimension log3(2), but the set of differences is the interval [ –1, 1], so that it does generate ℝ. It follows that the property under consideration is arithmetical rather than dimensional.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 31 , Issue 1 , June 1984 , pp. 42 - 46
Copyright
Copyright © University College London 1984

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References

1.Eggleston, H. G.. Sets of fractional dimensions which occur in some problems of number theory. Proc. London Math. Soc., 54 (1952–53), 42–93.CrossRefGoogle Scholar
2.Erdős, P. and Taylor, S. J.. On the set of points of convergence of a lacunary trigonometric series and the equidistribution properties of related sequences. Proc. London Math. Soc. (3), 7 (1957), 598–615.CrossRefGoogle Scholar
3.Connolly, T. H. and Williamson, J. H.. Difference-covers that are not k-sum-covers II. Proc. Camb. Phil. Soc., 75 (1974), 63–73.CrossRefGoogle Scholar
4.Haight, J. A.. Difference covers which have small k-sums for any k. Mathematika, 20 (1973), 109–118.CrossRefGoogle Scholar