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APPROXIMATING NUMBERS WITH MISSING DIGITS BY ALGEBRAIC NUMBERS

Published online by Cambridge University Press:  25 January 2007

Simon Kristensen
Simon Kristensen
Affiliation:
School of Mathematics, University of Edinburgh, James Clerk Maxwell Building, King’s Buildings, Mayfield Road, Edinburgh EH9 3JZ, UK
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Abstract

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We show that for a given base $b$ and a proper subset $E\subset\{0,\dots,b-1\}$, $\#E\ltb-1$, the set of numbers $x\in[0,1]$ that have no digits from $E$ in their expansion to base $b$ consists almost exclusively of $S^*$-numbers of type at most $\min\{2,\log b/\log(b-\#E)\}$. We also give upper bounds on the Hausdorff dimension of some exceptional sets.

Keywords

Primary 11J8311J82Diophantine approximationtranscendental numbersKoksma’s classificationCantor setsHausdorff dimension
Type
Research Article
Information
Proceedings of the Edinburgh Mathematical Society , Volume 49 , Issue 3 , October 2006 , pp. 657 - 666
Copyright
Copyright © Edinburgh Mathematical Society 2006