On the exceptional sets concerning the leading partial quotient in continued fractions. (English) Zbl 1468.11162
The present research is devoted to exceptional sets defined by the leading partial quotient in continued fractions.
Let \([a_1(x), a_2(x), \dots]\) be the continued fraction expansion of \(x \in (0, 1)\). Let \(T_n(x)\) the largest digit among the first \(n\) partial quotients of \(x\).
For any real numbers \(0 < \alpha < \beta < \infty\), the attention is given to the Hausdorff dimension of the exceptional set \[ F_{\phi}(\alpha, \beta)=\left\{x\in(0,1): \liminf_{n\to\infty}{\frac{T_n(x)}{\phi(n)}}=\alpha, ~ \limsup_{n\to\infty}{\frac{T_n(x)}{\phi(n)}}=\beta \right\}, \] where \(\phi(n)=n^a\) (here \(a>0\)) or \(\phi(n)=e^{n^{\gamma}}\) for \(\gamma>0\). It is proven that the Hausdorff dimension of the above exceptional set is full in the first case (i.e., for \(\phi(n)=n^a, a>0\)) and the Hausdorff dimension of \(F_{\phi}(\alpha, \beta)\) is equal to \(1\) or \(\frac 1 2\) in the second case.
Finally, some similar results on the exceptional sets for the sums of partial quotients in continued fractions are given.
Some elementary properties of continued fractions and auxiliary statements are given. The motivation of the present investigations is explained.
Let \([a_1(x), a_2(x), \dots]\) be the continued fraction expansion of \(x \in (0, 1)\). Let \(T_n(x)\) the largest digit among the first \(n\) partial quotients of \(x\).
For any real numbers \(0 < \alpha < \beta < \infty\), the attention is given to the Hausdorff dimension of the exceptional set \[ F_{\phi}(\alpha, \beta)=\left\{x\in(0,1): \liminf_{n\to\infty}{\frac{T_n(x)}{\phi(n)}}=\alpha, ~ \limsup_{n\to\infty}{\frac{T_n(x)}{\phi(n)}}=\beta \right\}, \] where \(\phi(n)=n^a\) (here \(a>0\)) or \(\phi(n)=e^{n^{\gamma}}\) for \(\gamma>0\). It is proven that the Hausdorff dimension of the above exceptional set is full in the first case (i.e., for \(\phi(n)=n^a, a>0\)) and the Hausdorff dimension of \(F_{\phi}(\alpha, \beta)\) is equal to \(1\) or \(\frac 1 2\) in the second case.
Finally, some similar results on the exceptional sets for the sums of partial quotients in continued fractions are given.
Some elementary properties of continued fractions and auxiliary statements are given. The motivation of the present investigations is explained.
Reviewer: Symon Serbenyuk (Kyïv)
MSC:
| 11K50 | Metric theory of continued fractions |
| 11K55 | Metric theory of other algorithms and expansions; measure and Hausdorff dimension |
| 28A80 | Fractals |
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