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A note on the approximation by continued fractions under an extra condition. (English) Zbl 0902.11031
Let \(p_n(x)/q_n(x)\) denote the convergents of any irrational number \(x\in[0,1]\) associated to the regular continued fraction expansion. For integers \(N\), \(a\), \(b\), \(m\), with \(1\leq a,b\leq m\), \(\text{gcd} (a,b,m)=1\), \(N>0\), and for \(t\geq 0\), the authors consider the number \(A(x,t,N; a,b)\) of integers \(n\) such that \(1\leq n\leq N\), \[ q_n(x)| q_n(x)x-p_n(x)| <t,\quad p_n(x)\equiv a\pmod m \quad\text{and}\quad q_n(x)\equiv b\pmod m. \] They prove that \(\lim_N N^{-1}A(x,t,N; a,b)=F(t)/J(m)\) almost everywhere (with respect to Lebesgue measure), where \(F(\cdot)\) is the Lenstra function and \(J(m)\) the Jordan’s totient function. This result follows from ergodic properties of the natural extension of a suitable skew product over the Gauss transformation, previously studied by H. Jager and P. Liardet [Indag. Math. 50, 181-197 (1988; Zbl 0665.10045)].
They also derive, for example, an asymptotic estimate of the number of couples \((p,q)\) of natural numbers satisfying \(\text{gcd} (p,q)=1\), \(p\equiv a \pmod m\), \(q\equiv b\pmod m\) and \(q| qx-p| <t\) with \(q<N\) and \(0\leq t\leq 1/2\). Similar results are obtained for \(S\)-expansions (including various continued fraction algorithms), considered by C. Kraaikamp [Acta Arith. 57, 1-39 (1991; Zbl 0721.11029)]. Also, a Gauss-Kusmin type theorem is obtained in the case of the regular continued fraction expansion.

11K50 Metric theory of continued fractions
28D20 Entropy and other invariants
11K55 Metric theory of other algorithms and expansions; measure and Hausdorff dimension
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