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.

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.

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