Irrationality Measure

Let be a real number, and let be the set of positive real numbers for which

(1)

has (at most) finitely many solutions for and integers. Then the irrationality measure, sometimes called the Liouville-Roth constant or irrationality exponent, is defined as the threshold at which Liouville's approximation theorem kicks in and is no longer approximable by rational numbers,

(2)

where is the infimum. If the set is empty, then is defined to be , and is called a Liouville number. There are three possible regimes for nonempty :

(3)

where the transitional case can correspond to being either algebraic of degree or being transcendental. Showing that for an algebraic number is a difficult result for which Roth was awarded the Fields medal.

The definition of irrationality measure is equivalent to the statement that if has irrationality measure , then is the smallest number such that the inequality

(4)

holds for any and all integers and with sufficiently large.

The irrationality measure of an irrational number can be given in terms of its simple continued fraction expansion and its convergents as

			(5)
			(6)

(Sondow 2004). For example, the golden ratio has

(7)

which follows immediately from (6) and the simple continued fraction expansion .

Exact values include for Liouville's constant and (Borwein and Borwein 1987, pp. 364-365). The best known upper bounds for other common constants as of mid-2020 are summarized in the following table, where is Apéry's constant, and are q-harmonic series, and the lower bounds are 2.

constant	upper bound	reference
	7.10320534	Zeilberger and Zudilin (2020)
	5.09541179	Zudilin (2013)
	3.57455391	Marcovecchio (2009)
	5.116201	Bondareva et al. (2018)
	5.513891	Rhin and Viola (2001)
	2.9384	Matala-Aho et al. (2006)
	2.4650	Zudilin (2004)

The bound for is due to Zeilberger and Zudilin (2020) and improves on the value 7.606308 previously found by Salikhov (2008). It has exact value given as follows. Let be the complex conjugate roots of

(8)

let be the positive real root, and let

			(9)
			(10)
			(11)
			(12)

then the bound is given by

(13)

Alekseyev (2011) has shown that the question of the convergence of the Flint Hills series is related to the irrationality measure of , and in particular, convergence would imply , which is much stronger than the best currently known upper bound.

An application outside number theory occurs in the semigroup stability of coupled partial differential equations. For a serially connected magnetizable piezoelectric-elastic beam model, the arithmetic nature of a quotient involving its physical parameters determines whether energy decay can be exponential or polynomial. In a class of irrational cases, the irrationality measure of this quotient gives an explicit polynomial decay exponent (Akil et al. 2024). This provides an example in which rational approximation properties of material-dependent parameters control the long-time behavior of a physical system.