Distribution of Diophantine approximation exponents for algebraic quantities in finite characteristic

Schmidt and Thakur proved that given any rational number μ between 2 and q+1, where q is a power of a prime p, there exists (explicitly given) algebraic Laurent series α in characteristic p, with their approximation exponents equal to μ and with degree of α being at most q+1. We first refine this result by showing that degree of α can be prescribed to be equal to q+1. Next we describe how the exponents of α 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 α have exponents near 2. We refine this result and give more precise descriptions of the distribution of the approximation exponents of such elements α of Class IA.