Uniformly Approximable Numbers and the Uniform Approximation Spectrum Original Research Article

We say a real numberαis uniformly approximable if the upper bound in Dirichletʹs theorem, from diophantine approximation, of 1/(Q+1) qmay be sharpened toc(α)/(Q+1)2for all sufficiently largeQ. 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 ofc(α), 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.