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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

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