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CMS Doctoral Prize / Prix de doctorat
(Organizers)
- STEPHEN ASTELS, University of Georgia, Athens, Georgia 30602, USA
Cantor sets and continued fractions
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Let C be the middle-third Cantor set constructed from the interval
[0,1]. Although C is rather sparse (the Hausdorff dimension of
C is log2 / log3), it can be show that C+C = [0,2], where we
are considering the point-wise sum of the sets. In this talk we will
examine sums, differences, products and quotients of more general types
of Cantor sets. We will apply these results to certain problems
arising in Diophantine approximation. For any set B of positive
integers define F(B) to be the set of numbers
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F(B) = {[t,a1,a2,...]; t Î Z,ai Î B for i ³ 1} |
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where [t,a1,a2,...] denotes the continued fraction t+1/(a1+1/(a2+\dotsm)). Let k be a positive integer and for
i = 1,...,k let Bi be a set of positive integers. We will give
conditions under which
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