Transcendency, measure of

2020 Mathematics Subject Classification: Primary: 11J82 [MSN][ZBL]

transcendence measure

A function characterizing the deviation of a given transcendental number from a set of algebraic numbers of bounded degree and bounded height under a change of bounds on these parameters. For a transcendental number $\omega$ and natural numbers $n$ and $H$, the measure of transcendency is

$$w_n(\omega;H)=\min|P(\omega)|,$$

where the minimum is taken over all non-zero integer polynomials of degree not exceeding $n$ and height not exceeding $H$. It follows from Dirichlet's "box" principle (cf. Dirichlet principle) that the following always holds:

$$w_n(\omega;H)<c_1^nH^{-n},$$

where $c_1$ depends only on $\omega$. In many cases it is possible to obtain not only a proof of the transcendency of a number $\omega$ but also a lower bound for the measure of transcendency in terms of the degree, and logarithmic or exponential functions of $n$ and $H$. For example, Hermite's method of proof of transcendency of $e$ enables one to obtain the inequality

$$w_n(e;h)>H^{-n-(c_2n^2\ln n)/\ln\ln H},$$

where $c_2>0$ is an absolute constant and $H\geq H_0(n)$. For any fixed $n$ and $\epsilon>0$,

$$w_n(\omega;H)>c_3H^{-n-\epsilon},\quad c_3=c_3(\omega;n,\epsilon)$$

for almost-all (in the sense of Lebesgue) real numbers $\omega$ (see Mahler problem). Transcendental numbers can be classified on the basis of the difference in asymptotic behaviour of $w_n(\omega;H)$ under unrestricted variation of $n$ and $H$ (see [3]).

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