Transcendency, measure of

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 and natural numbers and , the measure of transcendency is

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

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

where is an absolute constant and . For any fixed and ,

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

References

[1]	A.O. Gel'fond, "Transcendental and algebraic numbers" , Dover, reprint (1960) (Translated from Russian)
[2]	P.L. Cijsouw, "Transcendence measures" , Univ. Amsterdam (1972) (Dissertation)
[3]	A. Baker, "Transcendental number theory" , Cambridge Univ. Press (1975)

References

[a1]	A.B. Shidlovskii, "Transcendental numbers" , de Gruyter (1989) (Translated from Russian)