Distribution of Diophantine approximation exponents for algebraic quantities in finite characteristic

• Main Authors: Huei Jeng, 陳慧錚
• Other Authors: Liang-Chung Hsia
• Format: Others
• Language: en_US
• Published: 2011
• Online Access: http://ndltd.ncl.edu.tw/handle/27874139175297610540

博士 === 國立中央大學 === 數學研究所 === 99 === In contrast to Roth''s (Uchiyama''s respectively) theorem that algebraic real numbers (algebraic power series in characteristic zero respectively) have Diophantine approximation exponents equal to $2$, Mahler had shown that Liouville bound is the best possible in finite characteristic. Schmidt and Thakur proved that given any rational number $mu$ between $2$ and $q+1$, where $q$ is a power of a prime $p$, there exists (explicitly given) algebraic Laurent series $alpha$ in characteristic $p$, with their Diophantine approximation exponent equal to $mu$ and with degree of $alpha$ being at most $q+1$. We first refine this result by showing that degree of $alpha$ can be prescribed to be equal to $q+1$. Next we describe how the exponents of $alpha$''s are asymptotically distributed with respect to their heights in the case of algebraic elements of class IA for function fields over finite fields. A result of Thakur says that for low values of $q$ most elements $alpha$ have exponents near $2$. We refine this result and give more precise descriptions of the distribution of the approximation exponents of such elements $alpha$ of Class IA. In the last chapter, we compute the continued fractions and approximation exponents of certain families of elements related to Carlitz torsion.