Irrationality Measure

The irrationality measure of a real number is defined using the set of positive real numbers for which

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,

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:

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

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

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

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 and subsequent improvements 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.

Niedbala Giraudin (2026) proved that

where is a value of the Riemann zeta function.

Fauzan (2026) reports the upper bound

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

let be the positive real root, and let

then the bound is given by

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.

Casper (2026) reported that every nonzero rational number satisfies

where is the principal real inverse tangent. The argument extends OpenAI's interpolation and determinant method using Gaussian integers. The cases use the earlier claimed irrationality measure of . Casper credits OpenAI coding agents with substantive proof development and formalization. As of Oct. 25, 2026, the author reported Lean and exported-proof checks, but publication of the retained proof export awaited repository repairs and independent specialist review had not been reported.

Zhang and Li (2026) reported a quadratic field criterion for irrationality measure 2. Let be an integer that is not a square number, and let be a nonzero real number. Use the positive square root of for and for . If belongs to and has field norm 1, the claimed conclusion is that is an irrational number and . In particular, the criterion gives

for every positive rational number , and includes the same rational inverse tangent conclusion reported by Casper (2026). The argument uses the interpolation theorems of Liu et al. (2026) and OpenAI's (2026) determinant method. Zhang and Li report deriving the criterion themselves and credit GPT-6 Astra with assistance in exploring implications and constructing the Lean formalization. As of Nov. 3, 2026, independent specialist verification of the complete claimed proof had not been reported.

Leibtag et al. (2026) improved the bounds for Riemann zeta function values to

Since , the first bound also holds for . They additionally proved for every nonzero rational pair . Leibtag et al. (2026) report AI assistance with technical arithmetic estimates and formalization, while distinguishing their human-developed main independence construction.

For multiplicatively independent positive rational numbers , Chudnovsky (2026) reported

Multiplicative independence means for integers only when . In particular, the claimed bound applies to . Chudnovsky (2026) reports substantive ChatGPT assistance and a Lean formalization of this rational-logarithm-ratio theorem. That statement does not imply that every result in the broader manuscript has a complete formalization.