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On Hausdorff dimension of projections

Published online by Cambridge University Press:  26 February 2010

Robert Kaufman
Robert Kaufman
Affiliation:
University of Illinois, Urbana, Illinois.
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Let E be a compact subset of R2, of Hausdorff dimension s > 0 and for each real number t let Ft be the linear set {x1 + tx2: (x1x2) ∈ E}. In this note we shall prove

THEOREM. If s ≤ 1 then Ft has dimension ≥ s, excepting numbers t in a set of dimension ≤ s. If s > 1 then Ft, has positive Lebesgue measure, excepting numbers t in a set of Lebesgue measure 0.

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Type
Research Article
Information
Mathematika , Volume 15 , Issue 2 , December 1968, pp. 153 - 155
Copyright
Copyright © University College London 1968

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References

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