Irrationality measure

In mathematics, an irrationality measure of a real number

, takes positive real values and is strictly decreasing in both variables, consider the following inequality: for a given real number

,[1] a definition adapting the one of Liouville numbers — the irrationality exponent

is satisfied by an infinite number of coprime integer pairs

satisfying the above inequality yields good approximations of

, except for at most a finite number of "lucky" pairs

Rational numbers have irrationality exponent 1, while (as a consequence of Dirichlet's approximation theorem) every irrational number has irrationality exponent at least 2.

On the other hand, an application of Borel-Cantelli lemma shows that almost all numbers, including all algebraic irrational numbers, have an irrationality exponent exactly equal to 2.

, then we can establish an upper bound for the irrationality exponent of

by:[6][7] For most transcendental numbers, the exact value of their irrationality exponent is not known.

Examples include numbers which continued fractions behave predictably such as

[1][6] It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding

gives a stronger irrationality measure: the Markov constant

it is the factor by which Dirichlet's approximation theorem can be improved for

is a positive real number, then the inequality has infinitely many solutions

[33] This is in fact the best general lower bound since the golden ratio gives

by its simple continued fraction expansion, one may obtain:[34] Bounds for the Markov constant of

This gives rise to the question what the best upper bound is.

alongside the rationals and quadratic irrationals an exception to the fact that for almost all real numbers

:[5] (see Khinchin's theorem) Kurt Mahler extended the concept of an irrationality measure and defined a so-called transcendence measure, drawing on the idea of a Liouville number and partitioning the transcendental numbers into three distinct classes.

Mahler's irrationality measure can be generalized as follows:[2][3] Take

for which infinitely many such polynomials exist, that keep the inequality satisfied.

Then Mahler's transcendence measure is given by: The transcendental numbers can now be divided into the following three classes: If for all

In fact, almost all real numbers give

[38] They contain the Liouville numbers which are exactly the U-numbers of degree one.

Jurjen Koksma in 1939 proposed another generalization, similar to that of Mahler, based on approximations of complex numbers by algebraic numbers.

Then Koksma's transcendence measure is given by: The complex numbers can now once again be partitioned into four classes A*, S*, T* and U*.

However it turns out that these classes are equivalent to the ones given by Mahler in the sense that they produce exactly the same partition.

is taken to be an algebraic number that is also irrational one may obtain that the inequality has only at most finitely many solutions

one can quantify how well they can be approximated simultaneously by rational numbers