正特徵值函數體上的丟番圖逼近和有理數體以及零特徵值函數體上的丟番圖逼近不同, Mahler 舉出一個例子指出一個代數數逼近指數可以和它的擴張指數相同. Schmidt 和 Thakur 證明出, 給定任何一個介於 2和q+1的有理數m 我們都可以找到一組代數數使得它們的逼近指數等於m, 並且它的擴張指數比q+1小. 在此論文的第一部分中我們證明出了我們可以找出一組代數數使得它們的逼近指數等於m, 並且它的擴張指數等於q+1. 第二部分我們完整的描述了在
IA(q)的這個集合中的元素在區間 (2,q+1] 的逼近指數的分布. Thakur已經證明出在q小的時候大部分IA(q)的元素的逼近指數很接近2.
第三部分我們給出一些特殊代數數(由Carlitz 模來的) 的連分數公式以及逼近指數的計算. 第四部份我們給出了另一些特殊代數數(也是由Carlitz 模來的) 的逼近指數的上界.

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.

[1] W. Buck and D. Robbins. The continued fraction expansion of an
algebraic power series satisfying a quartic equation, J. Number Theory,
50: 335-344, 1995.
[2] L. E. Baum and M. M. Sweet. Continued fractions of algebraic power
series in characteristic 2. Ann. of Math. (2), 103(3):593{610, 1976.
[3] L. E. Baum and M. M. Sweet. Badly approximable power series in
characteristic 2. Ann. of Math. (2), 105(3):573{580, 1977.
[4] B. de Mathan. Approximations Diophantiennes dans un corps local.
Bull. Soc. Math. France Suppl. M em., 21:93, 1970.
[5] B. de Mathan. Approximation exponents for algebraic functions in
positive characteristic. Acta Arith., 60(4):359-370, 1992.
[6] B. de Mathan. Simultaneous Diophantine approximation for algebraic
functions in positive characteristic. Monatsh. Math. 111 (1991), no. 3,
187-193.
[7] B. de Mathan and O. Teuli e Problemes diophantiens simultanes.
(French) [Simultaneous Diophantine approximations] Monatsh. Math.
143 (2004), no. 3, 229-245.
[8] Neal Koblitz. p-adic numbers, p-adic analysis, and zeta-functions Graduate
Texts in Mathematics (v. 58)
[9] Kolchin E. Rational approximation to solutions of algebraic di erential
equations. Proc Amer Math Soc 10: 238244, 1959
[10] A. Lasjaunias , B. de Mathan Thue''s theorem in positive characteristic.
J Reine Angew Math 473: 195-206, 1996.
[11] A. Lasjaunias. A survey of Diophantine approximation in elds of power
series. Monatsh. Math., 130(3):211{229, 2000.
[12] A. Lasjaunias, W. Bluher Hyperquadratic power series of degree four
Acta Arithmetica, 124, 257-268, 2006.
[13] A. Lasjaunias. Continued fractions for hyperquadratic powerseries over
a nite eld. Finite elds and their applications, 14, 329-350, 2008.
[14] K. Mahler, On a theorem of Liouville in elds of positive charactersitic,
Canad. J. Math. 1 (1949), 397-400.
[15] W.H.Mills and David P. Robbins Continued fractions for certain algebraic
power series. J. Number Theory, 23: 388{404, 1986.
[16] C. Osgood. An e ective lower bound on the Diophantine approximation"
of algebraic functions by rational functions. Mathematika 20:
415, 1973
[17] C. Osgood. E ective bounds on the Diophantine approximation" of algebraic
functions over elds of arbitrary characteristic and applications
to di erential equations. Indag Math 37: 105-119, 1975.
[18] Michael Rosen. Number theory in function elds. Springer-Verlag New
York Berlin Heidelberg, Inc. 2002.
[19] K. F. Roth. Rational approximations to algebraic numbers. Mathe-
matika, 2:1{20; corrigendum, 168, 1955.
[20] W. M. Schmidt. On Osgood''s e ective Thue theorem for algebraic
functions. Comm Pure Applied Math 29: 759773,1976
[21] W. M. Schmidt. On continued fractions and Diophantine approximation
in power series elds. Acta Arith., 95(2):139{166, 2000.
[22] D. S. Thakur. Diophantine approximation exponents and continued
fractions for algebraic power series. J. Number Theory, 79(2):284{291,
1999.
[23] D. S. Thakur. Diophantine approximation and transcendence in -
nite characteristic In Diophantine Equations, Ed. N. Saradha, 265-278
(2008) Pub. for Tata Institute of Fundamental Research by Narosa Pub.
[24] D. S. Thakur. Function eld arithmetic. World Scienti c Publishing
Co. Inc., River Edge, NJ, 2004.
[25] D. S. Thakur. Approximation exponents for function elds Analytic
Number Theory- Essays in honor of Klaus Roth, Ed. W. Chen, T. Gowers,
H. Halberstram, W. Schmidt, R. Vaughn, Cambridge U. Press,
Cambridge, 421-435, 2009.
[26] A. Thue Om en generel i store hele tal ul sbar ligning. Videnskabs
Selskabets Skrifter 7: 1-15, 1908
[27] S. Uchiyama. On the Thue-Siegel-Roth theorem. III. Proc. Japan Acad.,
36:1{2, 1960.
[28] Voloch, J. F. Diophantine Approximation in Chracteristic p . Monatsh.
Math. 119 (1995), 321-325.