Distribution of Diophantine approximation exponents for algebraic quantities in finite characteristic

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• 作者：Huei-Jeng Chen
• 关键词：Diophantine approximation ; Finite characteristic ; Approximation exponents
• 刊名：Journal of Number Theory
• 出版年：2013
• 出版时间：November, 2013
• 年：2013
• 卷：133
• 期：11
• 页码：3620-3644
• 全文大小：322 K

文摘

Schmidt and Thakur proved that given any rational number ¦Ì between 2 and , where q is a power of a prime p, there exists (explicitly given) algebraic Laurent series ¦Á in characteristic p, with their approximation exponents equal to ¦Ì and with degree of ¦Á being at most . We first refine this result by showing that degree of ¦Á can be prescribed to be equal to . Next we describe how the exponents of ¦Á are asymptotically distributed with respect to their heights in the case of algebraic elements of Class IA for function fields over finite fields. Thakur had shown that most such elements ¦Á have exponents near 2. We refine this result and give more precise descriptions of the distribution of the approximation exponents of such elements ¦Á of Class IA.