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Uniformly Approximable Numbers and the Uniform Approximation Spectrum

โœ Scribed by Edward B. Burger

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
475 KB
Volume
61
Category
Article
ISSN
0022-314X

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โœฆ Synopsis

We say a real number : is uniformly approximable if the upper bound in Dirichlet's theorem, from diophantine approximation, of 1ร‚(Q+1) q may be sharpened to c(:)ร‚(Q+1) 2 for all sufficiently large Q. Here we begin by showing that the set of uniformly approximable numbers is precisely the set of badly approximable numbers. In additition, the optimal lower bound of c(:), referred to as the uniform approximation constant, is explicitly given. This allows us to introduce the notion of a uniform approximation spectrum. We conclude with a determination of the smallest values of this new spectrum and a comparison of this spectrum with other spectra.

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