The distribution of badly approximable rationals and continuants with bounded digits. II
[For Part I cf. C. R. Conf. Int., Québec/Can. 1987, 371-385 (1989; Zbl 0689.10042).] Theorems are proven on the density of ``badly approximable rationals'', that is on the relative frequency of the fractions a/b with \(1\leq a\leq b\leq x\), \((a,b)=1\) and, setting \(a/b=[0;a_ 1,...,a_ r]\), which denotes the finite regular continued fraction expansion of a/b, we have \(a_ k\leq n\) for \(k=1,.2,...,r\). The theorems establish the existence of distribution functions and contain statements about the behaviour of \[ Q_ n(x)=\sum_{1\leq a\leq b\leq x,a_ k<n,k<r}1\quad as\quad x\to \infty. \] Let us quote two of the results. Theorem 1. We have \(\lim_{x\to \infty}x^{-2E_ n}Q_ n(x)=C_ n,\) where \(E_ n\) is the Hausdorff dimension of the set \(t\in (0,1)\) whose partial quotients do not exceed n and \(C_ n\) is a constant depending only on n. Theorem 2. There is a continuous probability density function \(B_ n(\vartheta)\) such that \[ \lim_{x\to \infty}Q_ n(x)^{-1}\sum_{1\leq a\leq b\leq x,(a,b)=1,a| b\leq \vartheta,a_ k\leq n}1=B_ n(\vartheta). \]
distribution of badly approximable rationals
finite regular continued fraction expansion
scientific article; zbMATH DE number 4171022