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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.

MSC:
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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