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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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