Uniformly approximable numbers and the uniform approximation spectrum (Q678408)

Given an irrational number \(\alpha\), the uniform approximation constant \(\nu (\alpha)\) is defined as \[ \nu (\alpha)= \lim\sup_{Q\to \infty} \biggl((Q+1)^2 \min_{1\leq q\leq Q} \bigl\{|\alpha q |/q \bigr\} \biggr), \] where \(|\cdot|\) denotes the distance to the nearest integer. The author gives an explicit representation of \(\nu (\alpha)\) in terms of the simple continued fraction expansion of \(\alpha\). This is the starting point for a detailed investigation of the spectrum of values \(\nu (\alpha)\) and a comparison with the well-known Markov spectrum [cf. \textit{T. W. Cusick} and \textit{M. E. Flahive}, The Markov and Lagrange spectra, Math. Surveys Monographs 30 (1989; Zbl 0685.10023)] and with the dispersion spectrum of values \[ D(\alpha) =\lim\sup_{N \to\infty} N\sup_{0\leq x\leq 1} \min_{1\leq n\leq N} \Bigl\{\biggl |x-\bigl(n \alpha- [n\alpha] \bigr)\biggr |\Bigr\}, \] as introduced by E. Hlawka and later generalized by \textit{H. Niederreiter} [in Topics in classical number theory, Vol. II, Colloq. Math. Soc. Janos Bolyai 34, 1163-1208 (1984; Zbl 0547.10045)] (for more recent related work see [\textit{A. Tripathi}, Acta Arith. 63, 193-203 (1993; Zbl 0772.11023)]).