On the dispersion spectrum (Q1181788)

For irrational \(\alpha\) the dispersion constant \(D(\alpha)\) is defined as \(\lim\sup_{N\to \infty} Nd_ N\) with \(d_ N=\sup_{x\in[0,1]}\min_{1\leq n\leq N}| x-\{n\alpha\}|\) with \(\{ x\}\) being the fractional part of \(x\). It was shown by \textit{H. Niederreiter} [Topics in Classical Number Theory, Colloq. Math. Soc. Janos Bolyai 34, 1163-1208 (1984; Zbl 0547.10045)] that \(D(\alpha)<\infty\) iff \(\alpha\) has bounded partial quotients of its continued fraction expansion and the two smallest values of \(D(\alpha)\) were computed. The dispersion spectrum is the set of finite dispersion constants. The authors determine the smallest accumulation point \(\chi_ 1\) of the dispersion spectrum and all values \(\alpha\) and \(D(\alpha)\) for which \(D(\alpha)<\chi_ 1\).