Department of Mathematics, Columbia University, New York, NY 10027.
Proc Natl Acad Sci U S A. 1984 Mar;81(6):1926-30. doi: 10.1073/pnas.81.6.1926.
Siegel's results [Siegel, C. L. (1929) Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1] on the transcendence and algebraic independence of values of E-functions are refined to obtain the best possible bound for the measures of irrationality and linear independence of values of arbitrary E-functions at rational points. Our results show that values of E-functions at rational points have measures of diophantine approximations typical to "almost all" numbers. In particular, any such number has the "2 + epsilon" exponent of irrationality: Theta - p/q > q(-2-epsilon) for relatively prime rational integers p,q, with q >/= q(0) (Theta, epsilon). These results answer some problems posed by Lang. The methods used here are based on the introduction of graded Padé approximations to systems of functions satisfying linear differential equations with rational function coefficients. The constructions and proofs of this paper were used in the functional (nonarithmetic case) in a previous paper [Chudnovsky, D. V. & Chudnovsky, G. V. (1983) Proc. Natl. Acad. Sci. USA 80, 5158-5162].
西格尔的结果 [Siegel，C. L.（1929）Abh. Preuss. Akad. Wiss. Phys.-Math. Kl. 1] 关于 E 函数值的超越性和代数独立性被改进，以获得任意 E 函数在有理数点的值的无理性和线性独立性的最佳可能界。我们的结果表明，E 函数在有理数点的值具有典型的“几乎所有”数的丢番图逼近度量。特别地，任何这样的数都有“2 + epsilon”的无理数指数：对于互质的有理数整数 p，q，其中 q>=q(0)（Theta，epsilon），有 Theta - p/q > q(-2-epsilon)。这些结果回答了朗提出的一些问题。这里使用的方法基于引入满足有理函数系数线性微分方程的系统的分级 Padé 逼近。本文的构造和证明在以前的一篇论文 [Chudnovsky，D. V. & Chudnovsky，G. V.（1983）Proc. Natl. Acad. Sci. USA 80，5158-5162] 中用于泛函（非算术）情况。
