Hausdorff Dimensions Of Bounded-type Continued Fraction Sets Of Laurent Series

Based on the study of continued fractions and fractal geometry,this paper mainly studies the Hausdorff dimension of bounded-type continued fraction sets.We have known some properties of continued fraction sets and methods for calculating the Hausdorff dimension of a set whose partial quotient is bounded and continued fraction sets whose partial quotient approaches infinity at different speeds in the real field.And the set consisting of these dimensions is dense in the [0,1] interval in the real field.That is,the Texan conjecture is correct.Different from this,Based on the field of Laurent series.On the one hand,We research some properties of continued fraction sets in the field of Laurent series.For example,the length of the n order set and the regularized Haar measure defined on it are related to the degree of the former n partial quotients.On the other hand,We research the method of calculating the Hausdorff dimension of continued fraction sets whose degree of partial quotients is bounded in the field of Laurent series.The method of calculating dimension we used is called The principle of mass distribution.And whether or not the Texan conjecture is still established at this time,We have given the corresponding proof.The first part of the proof shows that the set consisting of those dimensions is dense in the interval[0,1/2],the second part shows that it is also dense in the interval[1/2,1].The conclusion of this paper lays a foundation for the property of continued fraction sets and Diophantine approximation on the field of formal Laurent series in the later study.It can also be applied to Lagrange spectra and Khintchine's theorem.Based on the study of this paper,we can also try to study the approximate range of the Hausdorff dimension of a finite continued fraction set whose partial quotients are fixed on two positive integers or the properties of other expansions in the field of formal series in the future,such as ?_expansions.