Irrationality Measure

The irrationality measure of a real number is defined using 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 the empty set, then is defined to be , and is called a Liouville number. There are three possible regimes for a nonempty set:

(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.

Equivalently, for irrational , is the supremum of the exponents for which rational approximations satisfying (2) occur infinitely often (Sondow 2004). Dirichlet's approximation theorem implies that for every irrational .

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 Minkowski-Bower constant also satisfies (Gayfulin and Shulga 2020). Selected upper bounds for other common constants are summarized in the following table, including the bound for reported by Fauzan (2026), 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)
	5.0193784	Niedbala Giraudin (2026)
	3.57455391	Marcovecchio (2009)
	5.116201	Bondareva et al. (2018)
	5.513891	Rhin and Viola (2001)
	260	Fauzan (2026), reported
	2.9384	Matala-Aho et al. (2006)
	2.4650	Zudilin (2004)

Niedbala Giraudin (2026) proved that

(8)

where is a value of the Riemann zeta function.

Fauzan (2026) reports the upper bound

(9)

where zeta(5) is a value of the Riemann zeta function. The bound follows from the estimate for every integer and all sufficiently large positive integer denominators .

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

(10)

let be the positive real root, and let

			(11)
			(12)
			(13)
			(14)

then the bound is given by

(15)

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. Meiburg (2022) proved that implies convergence of the Flint Hills series. The boundary case is not settled by these implications.

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.

OpenAI (2026) announced the exact value . This would mean that, for every , for every integer and all sufficiently large positive integer denominators . It does not assert a bound with fixed . The release attributes the argument to an unreleased internal OpenAI model and supplies a Lean formalization of the stated irrationality measure. As of Oct. 7, 2026, VibeMathed (2026) had read the formal statement but had not independently rebuilt it, and independent specialist verification of the complete proof had not been reported. The established upper bound above is separate from this claim.