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CMS Doctoral Prize / Prix de doctorat

STEPHEN ASTELS, University of Georgia, Athens, Georgia  30602, USA
Cantor sets and continued fractions

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

F(B) = {[t,a1,a2,...]; t Î Z,ai Î B for i ³ 1}
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
F(B1)±...±F(Bk) = R.

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