Rational Points in Cantor Sets
نویسندگان
چکیده
The Idea for this article was given by a problem in real analysis. We wanted to determine the one-dimensional Lebesgue-measure of the set f(C)9 where C stands for the classical triadic Cantor set and/is the Cantor-function, which is also known as "devil's staircase." We could see immediately that to determine the above measure we needed to know which dyadic rationals were contained in C. We soon found that the solution is well known; namely, there are only two such fractions: \ and j . This inspired a question: Are there any other primes such that only finitely many fractions are contained in the classical triadic Cantor set, where the denominator is a power of/?? The aim of this paper is to verify the surprising result: every p&3 prime fulfills the condition. Charles R. Wall showed in [2] that the Cantor set contains only 14 terminating decimals. His article gave very important information regarding the proof. We may ask if the quality of containing "very few" rational numbers and that of having zero Lebesgue measure are in close connection for a Cantor set. The answer seems to be "yes" at first sight, but in [1] Duane Boes, Richard Darst, and Paul Erdos showed a symmetric Cantor set family which, for each X e [0,1], has a member of Lebesgue measure 1-A, but the sets of the family typically do not contain "any" rational numbers.
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