Abstract
An estimate for the Hausdorff dimension of \({x\, \in \, \mathbb{R}}\) whose partial quotients of its regular continued fraction or minus continued fraction (MCF) are in \({E \, \subseteq \, \mathbb{N}}\) is given. This enables us to give a new proof for the Texan conjecture on \({[0,\,\frac{1}{2}]}\) which is valid for both regular and MCF. Also we show that if \({E \, \subseteq \,\mathbb{N}}\) and \({\sum_{e \, \in \, E} \,\frac{1}{e} \, = \, \infty}\) , then the Hausdorff dimension of E is at least \({\frac{1}{2}}\) .
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Dastjerdi, D.A., Lamei, S. Dimension of certain sets of regular and minus continued fractions with positive partial quotients. Arab. J. Math. 1, 139–148 (2012). https://doi.org/10.1007/s40065-012-0015-4
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DOI: https://doi.org/10.1007/s40065-012-0015-4