Irrationality measure

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Rational approximations to the square root of 2.

In mathematics, an irrationality measure of a real number x {\displaystyle x} {\displaystyle x} is a measure of how closely it can be approximated by a rational number.

If a function f ( t , λ ) {\displaystyle f(t,\lambda )} {\displaystyle f(t,\lambda )}, defined for t , λ > 0 {\displaystyle t,\lambda >0} {\displaystyle t,\lambda >0}, takes positive real values and is strictly decreasing in both variables, consider the following inequality:

0 < | x − p q | < f ( q , λ ) {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda )} {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda )}

for a given real number x ∈ R {\displaystyle x\in \mathbb {R} } {\displaystyle x\in \mathbb {R} } and rational numbers p q {\displaystyle {\frac {p}{q}}} {\displaystyle {\frac {p}{q}}} with p ∈ Z , q ∈ Z + {\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}} {\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}}. For x ∈ R {\displaystyle x\in \mathbb {R} } {\displaystyle x\in \mathbb {R} }, define R {\displaystyle R} {\displaystyle R} as the set of all λ ∈ R + {\displaystyle \lambda \in \mathbb {R} ^{+}} {\displaystyle \lambda \in \mathbb {R} ^{+}} for which the set { p q ∈ Q : | x − p q | < f ( q , λ ) } {\displaystyle \left\{{\frac {p}{q}}\in \mathbb {Q} :\left\vert x-{\frac {p}{q}}\right\vert <f(q,\lambda )\right\}} {\displaystyle \left\{{\frac {p}{q}}\in \mathbb {Q} :\left\vert x-{\frac {p}{q}}\right\vert <f(q,\lambda )\right\}} is finite. Then λ ( x ) = inf R {\displaystyle \lambda (x)=\inf R} {\displaystyle \lambda (x)=\inf R} is called an irrationality measure of x {\displaystyle x} {\displaystyle x} with regard to f . {\displaystyle f.} {\displaystyle f.} If the set R {\displaystyle R} {\displaystyle R} is empty, x {\displaystyle x} {\displaystyle x} is said to have infinite irrationality measure λ ( x ) = ∞ {\displaystyle \lambda (x)=\infty } {\displaystyle \lambda (x)=\infty }.

By definition, there is a sequence ( ε n ) n ∈ N {\displaystyle (\varepsilon _{n})_{n\in \mathbb {N} }} {\displaystyle (\varepsilon _{n})_{n\in \mathbb {N} }} converging to 0 such that for every n {\displaystyle n} {\displaystyle n}, the inequality

0 < | x − p q | < f ( q , λ ( x ) + ε n ) {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda (x)+\varepsilon _{n})} {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda (x)+\varepsilon _{n})}

has at most finitely many solutions p q {\displaystyle {\frac {p}{q}}} {\displaystyle {\frac {p}{q}}}. By the nondecreasing property of f {\displaystyle f} {\displaystyle f} in the second variable, we deduce that the inequality

0 < | x − p q | < f ( q , λ ( x ) + ε ) {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda (x)+\varepsilon )} {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<f(q,\lambda (x)+\varepsilon )}

has at most only finitely many solutions p q {\displaystyle {\frac {p}{q}}} {\displaystyle {\frac {p}{q}}} for all ε > 0 {\displaystyle \varepsilon >0} {\displaystyle \varepsilon >0}.[1]

Irrationality exponent

The irrationality exponent or Liouville–Roth irrationality measure is given by setting f ( q , μ ) = q − μ {\displaystyle f(q,\mu )=q^{-\mu }} {\displaystyle f(q,\mu )=q^{-\mu }},[1] a definition adapting the one of Liouville numbers — the irrationality exponent μ ( x ) {\displaystyle \mu (x)} {\displaystyle \mu (x)} is defined for real numbers x {\displaystyle x} {\displaystyle x} to be the supremum of the set of μ {\displaystyle \mu } {\displaystyle \mu } such that 0 < | x − p q | < 1 q μ {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}} {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}} is satisfied by an infinite number of coprime integer pairs ( p , q ) {\displaystyle (p,q)} {\displaystyle (p,q)} with q > 0 {\displaystyle q>0} {\displaystyle q>0}.[2][3]: 246 

For any value n < μ ( x ) {\displaystyle n<\mu (x)} {\displaystyle n<\mu (x)}, the infinite set of all rationals p / q {\displaystyle p/q} {\displaystyle p/q} satisfying the above inequality yields good approximations of x {\displaystyle x} {\displaystyle x}. Conversely, if n > μ ( x ) {\displaystyle n>\mu (x)} {\displaystyle n>\mu (x)}, then there are at most finitely many coprime ( p , q ) {\displaystyle (p,q)} {\displaystyle (p,q)} with q > 0 {\displaystyle q>0} {\displaystyle q>0} that satisfy the inequality.

For example, whenever a rational approximation p q ≈ x {\displaystyle {\frac {p}{q}}\approx x} {\displaystyle {\frac {p}{q}}\approx x} with p , q ∈ N {\displaystyle p,q\in \mathbb {N} } {\displaystyle p,q\in \mathbb {N} } yields n + 1 {\displaystyle n+1} {\displaystyle n+1} exact decimal digits, then

1 10 n ≥ | x − p q | ≥ 1 q μ ( x ) + ε {\displaystyle {\frac {1}{10^{n}}}\geq \left|x-{\frac {p}{q}}\right|\geq {\frac {1}{q^{\mu (x)+\varepsilon }}}} {\displaystyle {\frac {1}{10^{n}}}\geq \left|x-{\frac {p}{q}}\right|\geq {\frac {1}{q^{\mu (x)+\varepsilon }}}}

for any ε > 0 {\displaystyle \varepsilon >0} {\displaystyle \varepsilon >0}, except for at most a finite number of "lucky" pairs ( p , q ) {\displaystyle (p,q)} {\displaystyle (p,q)}.

A number x ∈ R {\displaystyle x\in \mathbb {R} } {\displaystyle x\in \mathbb {R} } with irrationality exponent μ ( x ) ≤ 2 {\displaystyle \mu (x)\leq 2} {\displaystyle \mu (x)\leq 2} is called a diophantine number,[4] while numbers with μ ( x ) = ∞ {\displaystyle \mu (x)=\infty } {\displaystyle \mu (x)=\infty } are called Liouville numbers.

Corollaries

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.[3]: 246 

It is μ ( x ) = μ ( r x + s ) {\displaystyle \mu (x)=\mu (rx+s)} {\displaystyle \mu (x)=\mu (rx+s)} for real numbers x {\displaystyle x} {\displaystyle x} and rational numbers r ≠ 0 {\displaystyle r\neq 0} {\displaystyle r\neq 0} and s {\displaystyle s} {\displaystyle s}. If for some x {\displaystyle x} {\displaystyle x} we have μ ( x ) ≤ μ {\displaystyle \mu (x)\leq \mu } {\displaystyle \mu (x)\leq \mu }, then it follows μ ( x 1 / 2 ) ≤ 2 μ {\displaystyle \mu (x^{1/2})\leq 2\mu } {\displaystyle \mu (x^{1/2})\leq 2\mu }.[5]: 368 

For a real number x {\displaystyle x} {\displaystyle x} given by its simple continued fraction expansion x = [ a 0 ; a 1 , a 2 , . . . ] {\displaystyle x=[a_{0};a_{1},a_{2},...]} {\displaystyle x=[a_{0};a_{1},a_{2},...]} with convergents p i / q i {\displaystyle p_{i}/q_{i}} {\displaystyle p_{i}/q_{i}} it holds:[1]

μ ( x ) = 1 + lim sup n → ∞ ln ⁡ q n + 1 ln ⁡ q n = 2 + lim sup n → ∞ ln ⁡ a n + 1 ln ⁡ q n . {\displaystyle \mu (x)=1+\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{\ln q_{n}}}=2+\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{\ln q_{n}}}.} {\displaystyle \mu (x)=1+\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{\ln q_{n}}}=2+\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{\ln q_{n}}}.}

If we have lim sup n → ∞ 1 n ln ⁡ | q n | ≤ σ {\displaystyle \limsup _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}|}\leq \sigma } {\displaystyle \limsup _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}|}\leq \sigma } and lim n → ∞ 1 n ln ⁡ | q n x − p n | = − τ {\displaystyle \lim _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}x-p_{n}|}=-\tau } {\displaystyle \lim _{n\to \infty }{\tfrac {1}{n}}{\ln |q_{n}x-p_{n}|}=-\tau } for some positive real numbers σ , τ {\displaystyle \sigma ,\tau } {\displaystyle \sigma ,\tau }, then we can establish an upper bound for the irrationality exponent of x {\displaystyle x} {\displaystyle x} by:[6][7]

μ ( x ) ≤ 1 + σ τ {\displaystyle \mu (x)\leq 1+{\frac {\sigma }{\tau }}} {\displaystyle \mu (x)\leq 1+{\frac {\sigma }{\tau }}}

Known bounds

For most transcendental numbers, the exact value of their irrationality exponent is not known.[5] Below is a table of known upper and lower bounds.

Number x {\displaystyle x} {\displaystyle x} Irrationality exponent μ ( x ) {\displaystyle \mu (x)} {\displaystyle \mu (x)} Notes
Lower bound Upper bound
Rational number p / q {\displaystyle p/q} {\displaystyle p/q} with p ∈ Z , q ∈ Z + {\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}} {\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}} 1 Every rational number p / q {\displaystyle p/q} {\displaystyle p/q} has an irrationality exponent of exactly 1.
Irrational algebraic number α {\displaystyle \alpha } {\displaystyle \alpha } 2 By Roth's theorem the irrationality exponent of any irrational algebraic number is exactly 2. Examples include square roots and the golden ratio φ {\displaystyle \varphi } {\displaystyle \varphi }.
e 2 / k , k ∈ Z + {\displaystyle e^{2/k},k\in \mathbb {Z} ^{+}} {\displaystyle e^{2/k},k\in \mathbb {Z} ^{+}} 2 If the elements a n {\displaystyle a_{n}} {\displaystyle a_{n}} of the simple continued fraction expansion of an irrational number x {\displaystyle x} {\displaystyle x} are bounded above a n < P ( n ) {\displaystyle a_{n}<P(n)} {\displaystyle a_{n}<P(n)} by an arbitrary polynomial P {\displaystyle P} {\displaystyle P}, then its irrationality exponent is μ ( x ) = 2 {\displaystyle \mu (x)=2} {\displaystyle \mu (x)=2}.

Examples include numbers whose continued fractions behave predictably such as

e = [ 2 ; 1 , 2 , 1 , 1 , 4 , 1 , 1 , 6 , 1 , . . . ] {\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...]} {\displaystyle e=[2;1,2,1,1,4,1,1,6,1,...]} and I 0 ( 2 ) / I 1 ( 2 ) = [ 1 ; 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , 10 , . . . ] {\displaystyle I_{0}(2)/I_{1}(2)=[1;2,3,4,5,6,7,8,9,10,...]} {\displaystyle I_{0}(2)/I_{1}(2)=[1;2,3,4,5,6,7,8,9,10,...]}.

tan ⁡ ( 1 / k ) , k ∈ Z + {\displaystyle \tan(1/k),k\in \mathbb {Z} ^{+}} {\displaystyle \tan(1/k),k\in \mathbb {Z} ^{+}} 2
tanh ⁡ ( 1 / k ) , k ∈ Z + {\displaystyle \tanh(1/k),k\in \mathbb {Z} ^{+}} {\displaystyle \tanh(1/k),k\in \mathbb {Z} ^{+}} 2
S ( b ) {\displaystyle S(b)} {\displaystyle S(b)} with b ≥ 2 {\displaystyle b\geq 2} {\displaystyle b\geq 2} 2 S ( b ) := ∑ k = 0 ∞ b − 2 k {\displaystyle S(b):=\sum _{k=0}^{\infty }b^{-2^{k}}} {\displaystyle S(b):=\sum _{k=0}^{\infty }b^{-2^{k}}}with b ∈ Z {\displaystyle b\in \mathbb {Z} } {\displaystyle b\in \mathbb {Z} }, has continued fraction terms which do not exceed a fixed constant.[8][9]
T ( b ) {\displaystyle T(b)} {\displaystyle T(b)} with b ≥ 2 {\displaystyle b\geq 2} {\displaystyle b\geq 2}[10] 2 T ( b ) := ∑ k = 0 ∞ t k b − k {\displaystyle T(b):=\sum _{k=0}^{\infty }t_{k}b^{-k}} {\displaystyle T(b):=\sum _{k=0}^{\infty }t_{k}b^{-k}} where t k {\displaystyle t_{k}} {\displaystyle t_{k}} is the Thue–Morse sequence and b ∈ Z {\displaystyle b\in \mathbb {Z} } {\displaystyle b\in \mathbb {Z} }. See Prouhet-Thue-Morse constant.
ln ⁡ ( 2 ) {\displaystyle \ln(2)} {\displaystyle \ln(2)}[11][12] 2 3.57455... There are other numbers of the form ln ⁡ ( a / b ) {\displaystyle \ln(a/b)} {\displaystyle \ln(a/b)} for which bounds on their irrationality exponents are known.[13][14][15]
ln ⁡ ( 3 ) {\displaystyle \ln(3)} {\displaystyle \ln(3)}[11][16] 2 5.11620...
5 ln ⁡ ( 3 / 2 ) {\displaystyle 5\ln(3/2)} {\displaystyle 5\ln(3/2)}[17] 2 3.43506... There are many other numbers of the form 2 k + 1 ln ⁡ ( 2 k + 1 + 1 2 k + 1 − 1 ) {\displaystyle {\sqrt {2k+1}}\ln \left({\frac {{\sqrt {2k+1}}+1}{{\sqrt {2k+1}}-1}}\right)} {\displaystyle {\sqrt {2k+1}}\ln \left({\frac {{\sqrt {2k+1}}+1}{{\sqrt {2k+1}}-1}}\right)} for which bounds on their irrationality exponents are known.[17] This is the case for k = 12 {\displaystyle k=12} {\displaystyle k=12}.
π / 3 {\displaystyle \pi /{\sqrt {3}}} {\displaystyle \pi /{\sqrt {3}}}[18][19] 2 4.60105... There are many other numbers of the form 2 k − 1 arctan ⁡ ( 2 k − 1 k − 1 ) {\displaystyle {\sqrt {2k-1}}\arctan \left({\frac {\sqrt {2k-1}}{k-1}}\right)} {\displaystyle {\sqrt {2k-1}}\arctan \left({\frac {\sqrt {2k-1}}{k-1}}\right)} for which bounds on their irrationality exponents are known.[18] This is the case for k = 2 {\displaystyle k=2} {\displaystyle k=2}.
π {\displaystyle \pi } {\displaystyle \pi }[11][20] 2 7.10320... It has been proven that π {\displaystyle \pi } {\displaystyle \pi }'s irrationality exponent is at most 5 / 2 {\displaystyle 5/2} {\displaystyle 5/2} if the Flint Hills series ∑ n = 1 ∞ csc 2 ⁡ n n 3 {\displaystyle \displaystyle \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}}} {\displaystyle \displaystyle \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}}} (where n is in radians) converges,[21][22] and that if the irrationality exponent is at least 5 / 2 {\displaystyle 5/2} {\displaystyle 5/2} it diverges.[23] Conversely,[23] (theorem 2.5) proved that the Flint Hills series converges if π {\displaystyle \pi } {\displaystyle \pi }'s irrationality exponent is ≤ 5 / 2 {\displaystyle \leq 5/2} {\displaystyle \leq 5/2}, completing the biconditional (unless μ ( π ) = 5 / 2 {\displaystyle \mu (\pi )=5/2} {\displaystyle \mu (\pi )=5/2}). In 2026, OpenAI released an unverified proof claiming that μ ( π ) = 2 {\displaystyle \mu (\pi )=2} {\displaystyle \mu (\pi )=2}.[24]
π 2 {\displaystyle \pi ^{2}} {\displaystyle \pi ^{2}}[11][25] 2 5.09541... π 2 {\displaystyle \pi ^{2}} {\displaystyle \pi ^{2}} and ζ ( 2 ) {\displaystyle \zeta (2)} {\displaystyle \zeta (2)} are linearly dependent over Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} }. ( ζ ( 2 ) = π 2 6 ) {\displaystyle \left(\zeta (2)={\frac {\pi ^{2}}{6}}\right)} {\displaystyle \left(\zeta (2)={\frac {\pi ^{2}}{6}}\right)}, also see the Basel problem.
arctan ⁡ ( 1 / 2 ) {\displaystyle \arctan(1/2)} {\displaystyle \arctan(1/2)}[26] 2 9.27204... There are many other numbers of the form arctan ⁡ ( 1 / k ) {\displaystyle \arctan(1/k)} {\displaystyle \arctan(1/k)} for which bounds on their irrationality exponents are known.[27][28]
arctan ⁡ ( 1 / 3 ) {\displaystyle \arctan(1/3)} {\displaystyle \arctan(1/3)}[29] 2 5.94202...
Apéry's constant ζ ( 3 ) {\displaystyle \zeta (3)} {\displaystyle \zeta (3)}[11] 2 5.51389...
Γ ( 1 / 4 ) {\displaystyle \Gamma (1/4)} {\displaystyle \Gamma (1/4)}[30] 2 10143
Cahen's constant C {\displaystyle C} {\displaystyle C}[31] 3
Champernowne constants C b {\displaystyle C_{b}} {\displaystyle C_{b}} in base b ≥ 2 {\displaystyle b\geq 2} {\displaystyle b\geq 2}[32] b {\displaystyle b} {\displaystyle b} Examples include C 10 = 0.1234567891011... = [ 0 ; 8 , 9 , 1 , 149083 , 1 , . . . ] {\displaystyle C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]} {\displaystyle C_{10}=0.1234567891011...=[0;8,9,1,149083,1,...]}
Liouville numbers L {\displaystyle L} {\displaystyle L} ∞ {\displaystyle \infty } {\displaystyle \infty } The Liouville numbers are precisely those numbers having infinite irrationality exponent.[3]: 248 

Irrationality base

The irrationality base or Sondow irrationality measure is obtained by setting f ( q , β ) = β − q {\displaystyle f(q,\beta )=\beta ^{-q}} {\displaystyle f(q,\beta )=\beta ^{-q}}.[1][6] It is a weaker irrationality measure, being able to distinguish how well different Liouville numbers can be approximated, but yielding β ( x ) = 1 {\displaystyle \beta (x)=1} {\displaystyle \beta (x)=1} for all other real numbers:

Let x {\displaystyle x} {\displaystyle x} be an irrational number. If there exist real numbers β ≥ 1 {\displaystyle \beta \geq 1} {\displaystyle \beta \geq 1} with the property that for any ε > 0 {\displaystyle \varepsilon >0} {\displaystyle \varepsilon >0}, there is a positive integer q ( ε ) {\displaystyle q(\varepsilon )} {\displaystyle q(\varepsilon )} such that

| x − p q | > 1 ( β + ε ) q {\displaystyle \left|x-{\frac {p}{q}}\right|>{\frac {1}{(\beta +\varepsilon )^{q}}}} {\displaystyle \left|x-{\frac {p}{q}}\right|>{\frac {1}{(\beta +\varepsilon )^{q}}}}

for all integers p , q {\displaystyle p,q} {\displaystyle p,q} with q ≥ q ( ε ) {\displaystyle q\geq q(\varepsilon )} {\displaystyle q\geq q(\varepsilon )} then the least such β {\displaystyle \beta } {\displaystyle \beta } is called the irrationality base of x {\displaystyle x} {\displaystyle x} and is represented as β ( x ) {\displaystyle \beta (x)} {\displaystyle \beta (x)}.

If no such β {\displaystyle \beta } {\displaystyle \beta } exists, then β ( x ) = ∞ {\displaystyle \beta (x)=\infty } {\displaystyle \beta (x)=\infty } and x {\displaystyle x} {\displaystyle x} is called a super Liouville number.

If a real number x {\displaystyle x} {\displaystyle x} is given by its simple continued fraction expansion x = [ a 0 ; a 1 , a 2 , . . . ] {\displaystyle x=[a_{0};a_{1},a_{2},...]} {\displaystyle x=[a_{0};a_{1},a_{2},...]} with convergents p i / q i {\displaystyle p_{i}/q_{i}} {\displaystyle p_{i}/q_{i}} then it holds:

β ( x ) = lim sup n → ∞ ln ⁡ q n + 1 q n = lim sup n → ∞ ln ⁡ a n + 1 q n {\displaystyle \beta (x)=\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{q_{n}}}=\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{q_{n}}}} {\displaystyle \beta (x)=\limsup _{n\to \infty }{\frac {\ln q_{n+1}}{q_{n}}}=\limsup _{n\to \infty }{\frac {\ln a_{n+1}}{q_{n}}}}.[1]

Examples

Any real number x {\displaystyle x} {\displaystyle x} with finite irrationality exponent μ ( x ) < ∞ {\displaystyle \mu (x)<\infty } {\displaystyle \mu (x)<\infty } has irrationality base β ( x ) = 1 {\displaystyle \beta (x)=1} {\displaystyle \beta (x)=1}, while any number with irrationality base β ( x ) > 1 {\displaystyle \beta (x)>1} {\displaystyle \beta (x)>1} has irrationality exponent μ ( x ) = ∞ {\displaystyle \mu (x)=\infty } {\displaystyle \mu (x)=\infty } and is a Liouville number.

The number L = [ 1 ; 2 , 2 2 , 2 2 2 , . . . ] {\displaystyle L=[1;2,2^{2},2^{2^{2}},...]} {\displaystyle L=[1;2,2^{2},2^{2^{2}},...]} has irrationality exponent μ ( L ) = ∞ {\displaystyle \mu (L)=\infty } {\displaystyle \mu (L)=\infty } and irrationality base β ( L ) = 1 {\displaystyle \beta (L)=1} {\displaystyle \beta (L)=1}.

The numbers τ a = ∑ n = 0 ∞ 1 n a = 1 + 1 a + 1 a a + 1 a a a + 1 a a a a + . . . {\displaystyle \tau _{a}=\sum _{n=0}^{\infty }{\frac {1}{^{n}a}}=1+{\frac {1}{a}}+{\frac {1}{a^{a}}}+{\frac {1}{a^{a^{a}}}}+{\frac {1}{a^{a^{a^{a}}}}}+...} {\displaystyle \tau _{a}=\sum _{n=0}^{\infty }{\frac {1}{^{n}a}}=1+{\frac {1}{a}}+{\frac {1}{a^{a}}}+{\frac {1}{a^{a^{a}}}}+{\frac {1}{a^{a^{a^{a}}}}}+...} ( n a {\displaystyle {^{n}a}} {\displaystyle {^{n}a}} represents tetration, a = 2 , 3 , 4... {\displaystyle a=2,3,4...} {\displaystyle a=2,3,4...}) have irrationality base β ( τ a ) = a {\displaystyle \beta (\tau _{a})=a} {\displaystyle \beta (\tau _{a})=a}.

The number S = 1 + 1 2 1 + 1 4 2 1 + 1 8 4 2 1 + 1 16 8 4 2 1 + 1 32 16 8 4 2 1 + … {\displaystyle S=1+{\frac {1}{2^{1}}}+{\frac {1}{4^{2^{1}}}}+{\frac {1}{8^{4^{2^{1}}}}}+{\frac {1}{16^{8^{4^{2^{1}}}}}}+{\frac {1}{32^{16^{8^{4^{2^{1}}}}}}}+\ldots } {\displaystyle S=1+{\frac {1}{2^{1}}}+{\frac {1}{4^{2^{1}}}}+{\frac {1}{8^{4^{2^{1}}}}}+{\frac {1}{16^{8^{4^{2^{1}}}}}}+{\frac {1}{32^{16^{8^{4^{2^{1}}}}}}}+\ldots } has irrationality base β ( S ) = ∞ {\displaystyle \beta (S)=\infty } {\displaystyle \beta (S)=\infty }, hence it is a super Liouville number.

Although it is not known whether or not e π {\displaystyle e^{\pi }} {\displaystyle e^{\pi }} is a Liouville number,[33]: 20  it is known that β ( e π ) = 1 {\displaystyle \beta (e^{\pi })=1} {\displaystyle \beta (e^{\pi })=1}.[5]: 371 

Other irrationality measures

Markov constant

Setting f ( q , M ) = ( M q 2 ) − 1 {\displaystyle f(q,M)=(Mq^{2})^{-1}} {\displaystyle f(q,M)=(Mq^{2})^{-1}} gives a stronger irrationality measure: the Markov constant M ( x ) {\displaystyle M(x)} {\displaystyle M(x)}. For an irrational number x ∈ R ∖ Q {\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} } {\displaystyle x\in \mathbb {R} \setminus \mathbb {Q} } it is the factor by which Dirichlet's approximation theorem can be improved for x {\displaystyle x} {\displaystyle x}. Namely if c < M ( x ) {\displaystyle c<M(x)} {\displaystyle c<M(x)} is a positive real number, then the inequality

0 < | x − p q | < 1 c q 2 {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}} {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}}

has infinitely many solutions p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } {\displaystyle {\frac {p}{q}}\in \mathbb {Q} }. If c > M ( x ) {\displaystyle c>M(x)} {\displaystyle c>M(x)} there are at most finitely many solutions.

Dirichlet's approximation theorem implies M ( x ) ≥ 1 {\displaystyle M(x)\geq 1} {\displaystyle M(x)\geq 1} and Hurwitz's theorem gives M ( x ) ≥ 5 {\displaystyle M(x)\geq {\sqrt {5}}} {\displaystyle M(x)\geq {\sqrt {5}}} both for irrational x {\displaystyle x} {\displaystyle x}.[34]

This is in fact the best general lower bound since the golden ratio gives M ( φ ) = 5 {\displaystyle M(\varphi )={\sqrt {5}}} {\displaystyle M(\varphi )={\sqrt {5}}}. It is also M ( 2 ) = 2 2 {\displaystyle M({\sqrt {2}})=2{\sqrt {2}}} {\displaystyle M({\sqrt {2}})=2{\sqrt {2}}}.

Given x = [ a 0 ; a 1 , a 2 , . . . ] {\displaystyle x=[a_{0};a_{1},a_{2},...]} {\displaystyle x=[a_{0};a_{1},a_{2},...]} by its simple continued fraction expansion, one may obtain:[35]

M ( x ) = lim sup n → ∞ ( [ a n + 1 ; a n + 2 , a n + 3 , . . . ] + [ 0 ; a n , a n − 1 , . . . , a 2 , a 1 ] ) . {\displaystyle M(x)=\limsup _{n\to \infty }{([a_{n+1};a_{n+2},a_{n+3},...]+[0;a_{n},a_{n-1},...,a_{2},a_{1}])}.} {\displaystyle M(x)=\limsup _{n\to \infty }{([a_{n+1};a_{n+2},a_{n+3},...]+[0;a_{n},a_{n-1},...,a_{2},a_{1}])}.}

Bounds for the Markov constant of x = [ a 0 ; a 1 , a 2 , . . . ] {\displaystyle x=[a_{0};a_{1},a_{2},...]} {\displaystyle x=[a_{0};a_{1},a_{2},...]} can also be given by p 2 + 4 ≤ M ( x ) < p + 2 {\displaystyle {\sqrt {p^{2}+4}}\leq M(x)<p+2} {\displaystyle {\sqrt {p^{2}+4}}\leq M(x)<p+2} with p = lim sup n → ∞ a n {\displaystyle p=\limsup _{n\to \infty }a_{n}} {\displaystyle p=\limsup _{n\to \infty }a_{n}}.[36] This implies that M ( x ) = ∞ {\displaystyle M(x)=\infty } {\displaystyle M(x)=\infty } if and only if ( a k ) {\displaystyle (a_{k})} {\displaystyle (a_{k})} is not bounded. The numbers with finite M ( x ) {\displaystyle M(x)} {\displaystyle M(x)} are called badly approximable. In particular every quadratic irrational number x {\displaystyle x} {\displaystyle x} is badly approximable.

Any number with μ ( x ) > 2 {\displaystyle \mu (x)>2} {\displaystyle \mu (x)>2} or β ( x ) > 1 {\displaystyle \beta (x)>1} {\displaystyle \beta (x)>1} has an unbounded simple continued fraction and hence M ( x ) = ∞ {\displaystyle M(x)=\infty } {\displaystyle M(x)=\infty }. A further consequence is M ( e ) = ∞ {\displaystyle M(e)=\infty } {\displaystyle M(e)=\infty }.

For rational numbers r {\displaystyle r} {\displaystyle r} it may be defined M ( r ) = 0 {\displaystyle M(r)=0} {\displaystyle M(r)=0}.

Other results

The values M ( e ) = ∞ {\displaystyle M(e)=\infty } {\displaystyle M(e)=\infty } and μ ( e ) = 2 {\displaystyle \mu (e)=2} {\displaystyle \mu (e)=2} imply that the inequality 0 < | e − p q | < 1 c q 2 {\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}} {\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{cq^{2}}}} has for all c ∈ R + {\displaystyle c\in \mathbb {R} ^{+}} {\displaystyle c\in \mathbb {R} ^{+}} infinitely many solutions p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } while the inequality 0 < | e − p q | < 1 q 2 + ε {\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{q^{2+\varepsilon }}}} {\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {1}{q^{2+\varepsilon }}}} has for all ε ∈ R + {\displaystyle \varepsilon \in \mathbb {R} ^{+}} {\displaystyle \varepsilon \in \mathbb {R} ^{+}} only at most finitely many solutions p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } {\displaystyle {\frac {p}{q}}\in \mathbb {Q} }. This gives rise to the question what the best upper bound is. The answer is given by:[37]

0 < | e − p q | < c ln ⁡ ln ⁡ q q 2 ln ⁡ q {\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {c\ln \ln q}{q^{2}\ln q}}} {\displaystyle 0<\left|e-{\frac {p}{q}}\right|<{\frac {c\ln \ln q}{q^{2}\ln q}}}

which is satisfied by infinitely many p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } for c > 1 2 {\displaystyle c>{\tfrac {1}{2}}} {\displaystyle c>{\tfrac {1}{2}}} but not for c < 1 2 {\displaystyle c<{\tfrac {1}{2}}} {\displaystyle c<{\tfrac {1}{2}}}.

This makes the number e {\displaystyle e} {\displaystyle e} alongside the rationals and quadratic irrationals an exception to the fact that for almost all real numbers x ∈ R {\displaystyle x\in \mathbb {R} } {\displaystyle x\in \mathbb {R} } the inequality below has infinitely many solutions p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } {\displaystyle {\frac {p}{q}}\in \mathbb {Q} }:[5] (see Khinchin's theorem)

0 < | x − p q | < 1 q 2 ln ⁡ q {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{2}\ln q}}} {\displaystyle 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{2}\ln q}}}

Mahler's generalization

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.[3]

Mahler's irrationality measure

Instead of taking for a given real number x {\displaystyle x} {\displaystyle x} the difference | x − p / q | {\displaystyle |x-p/q|} {\displaystyle |x-p/q|} with p / q ∈ Q {\displaystyle p/q\in \mathbb {Q} } {\displaystyle p/q\in \mathbb {Q} }, one may instead focus on term | q x − p | = | L ( x ) | {\displaystyle |qx-p|=|L(x)|} {\displaystyle |qx-p|=|L(x)|} with p , q ∈ Z {\displaystyle p,q\in \mathbb {Z} } {\displaystyle p,q\in \mathbb {Z} } and L ∈ Z [ x ] {\displaystyle L\in \mathbb {Z} [x]} {\displaystyle L\in \mathbb {Z} [x]} with deg ⁡ L = 1 {\displaystyle \deg L=1} {\displaystyle \deg L=1}. Consider the following inequality:

0 < | q x − p | ≤ max ( | p | , | q | ) − ω {\displaystyle 0<|qx-p|\leq \max(|p|,|q|)^{-\omega }} {\displaystyle 0<|qx-p|\leq \max(|p|,|q|)^{-\omega }} with p , q ∈ Z {\displaystyle p,q\in \mathbb {Z} } {\displaystyle p,q\in \mathbb {Z} } and ω ∈ R 0 + {\displaystyle \omega \in \mathbb {R} _{0}^{+}} {\displaystyle \omega \in \mathbb {R} _{0}^{+}}.

Define R {\displaystyle R} {\displaystyle R} as the set of all ω ∈ R 0 + {\displaystyle \omega \in \mathbb {R} _{0}^{+}} {\displaystyle \omega \in \mathbb {R} _{0}^{+}} for which infinitely many solutions p , q ∈ Z {\displaystyle p,q\in \mathbb {Z} } {\displaystyle p,q\in \mathbb {Z} } exist, such that the inequality is satisfied. Then ω 1 ( x ) = sup M {\displaystyle \omega _{1}(x)=\sup M} {\displaystyle \omega _{1}(x)=\sup M} is Mahler's irrationality measure. It gives ω 1 ( p / q ) = 0 {\displaystyle \omega _{1}(p/q)=0} {\displaystyle \omega _{1}(p/q)=0} for rational numbers, ω 1 ( α ) = 1 {\displaystyle \omega _{1}(\alpha )=1} {\displaystyle \omega _{1}(\alpha )=1} for algebraic irrational numbers and in general ω 1 ( x ) = μ ( x ) − 1 {\displaystyle \omega _{1}(x)=\mu (x)-1} {\displaystyle \omega _{1}(x)=\mu (x)-1}, where μ ( x ) {\displaystyle \mu (x)} {\displaystyle \mu (x)} denotes the irrationality exponent.

Transcendence measure

Mahler's irrationality measure can be generalized as follows:[2][3] Take P {\displaystyle P} {\displaystyle P} to be a polynomial with deg ⁡ P ≤ n ∈ Z + {\displaystyle \deg P\leq n\in \mathbb {Z} ^{+}} {\displaystyle \deg P\leq n\in \mathbb {Z} ^{+}} and integer coefficients a i ∈ Z {\displaystyle a_{i}\in \mathbb {Z} } {\displaystyle a_{i}\in \mathbb {Z} }. Then define a height function H ( P ) = max ( | a 0 | , | a 1 | , . . . , | a n | ) {\displaystyle H(P)=\max(|a_{0}|,|a_{1}|,...,|a_{n}|)} {\displaystyle H(P)=\max(|a_{0}|,|a_{1}|,...,|a_{n}|)} and consider for complex numbers z {\displaystyle z} {\displaystyle z} the inequality:

0 < | P ( z ) | ≤ H ( P ) − ω {\displaystyle 0<|P(z)|\leq H(P)^{-\omega }} {\displaystyle 0<|P(z)|\leq H(P)^{-\omega }} with ω ∈ R 0 + {\displaystyle \omega \in \mathbb {R} _{0}^{+}} {\displaystyle \omega \in \mathbb {R} _{0}^{+}}.

Set R {\displaystyle R} {\displaystyle R} to be the set of all ω ∈ R 0 + {\displaystyle \omega \in \mathbb {R} _{0}^{+}} {\displaystyle \omega \in \mathbb {R} _{0}^{+}} for which infinitely many such polynomials exist, that keep the inequality satisfied. Further define ω n ( z ) = sup R {\displaystyle \omega _{n}(z)=\sup R} {\displaystyle \omega _{n}(z)=\sup R} for all n ∈ Z + {\displaystyle n\in \mathbb {Z} ^{+}} {\displaystyle n\in \mathbb {Z} ^{+}} with ω 1 ( z ) {\displaystyle \omega _{1}(z)} {\displaystyle \omega _{1}(z)} being the above irrationality measure, ω 2 ( z ) {\displaystyle \omega _{2}(z)} {\displaystyle \omega _{2}(z)} being a non-quadraticity measure, etc.

Then Mahler's transcendence measure is given by:

ω ( z ) = lim sup n → ∞ ω n ( z ) . {\displaystyle \omega (z)=\limsup _{n\to \infty }\omega _{n}(z).} {\displaystyle \omega (z)=\limsup _{n\to \infty }\omega _{n}(z).}

The transcendental numbers can now be divided into the following three classes:

If for all n ∈ Z + {\displaystyle n\in \mathbb {Z} ^{+}} {\displaystyle n\in \mathbb {Z} ^{+}} the value of ω n ( z ) {\displaystyle \omega _{n}(z)} {\displaystyle \omega _{n}(z)} is finite and ω ( z ) {\displaystyle \omega (z)} {\displaystyle \omega (z)} is finite as well, z {\displaystyle z} {\displaystyle z} is called an S-number (of type ω ( z ) {\displaystyle \omega (z)} {\displaystyle \omega (z)}).

If for all n ∈ Z + {\displaystyle n\in \mathbb {Z} ^{+}} {\displaystyle n\in \mathbb {Z} ^{+}} the value of ω n ( z ) {\displaystyle \omega _{n}(z)} {\displaystyle \omega _{n}(z)} is finite but ω ( z ) {\displaystyle \omega (z)} {\displaystyle \omega (z)} is infinite, z {\displaystyle z} {\displaystyle z} is called an T-number.

If there exists a smallest positive integer N {\displaystyle N} {\displaystyle N} such that for all n ≥ N {\displaystyle n\geq N} {\displaystyle n\geq N} the ω n ( z ) {\displaystyle \omega _{n}(z)} {\displaystyle \omega _{n}(z)} are infinite, z {\displaystyle z} {\displaystyle z} is called an U-number (of degree N {\displaystyle N} {\displaystyle N}).

The number z {\displaystyle z} {\displaystyle z} is algebraic (and called an A-number) if and only if ω ( z ) = 0 {\displaystyle \omega (z)=0} {\displaystyle \omega (z)=0}.

Almost all numbers are S-numbers. In fact, almost all real numbers give ω ( x ) = 1 {\displaystyle \omega (x)=1} {\displaystyle \omega (x)=1} while almost all complex numbers give ω ( z ) = 1 2 {\displaystyle \omega (z)={\tfrac {1}{2}}} {\displaystyle \omega (z)={\tfrac {1}{2}}}.[38]: 86  The number e is an S-number with ω ( e ) = 1 {\displaystyle \omega (e)=1} {\displaystyle \omega (e)=1}. The number π is either an S- or T-number.[38]: 86  The U-numbers are a set of measure 0 but still uncountable.[39] They contain the Liouville numbers which are exactly the U-numbers of degree one.

Linear independence measure

Another generalization of Mahler's irrationality measure gives a linear independence measure.[2][13] For real numbers x 1 , . . . , x n ∈ R {\displaystyle x_{1},...,x_{n}\in \mathbb {R} } {\displaystyle x_{1},...,x_{n}\in \mathbb {R} } consider the inequality

0 < | c 1 x 1 + . . . + c n x n | ≤ max ( | c 1 | , . . . , | c n | ) − ν {\displaystyle 0<|c_{1}x_{1}+...+c_{n}x_{n}|\leq \max(|c_{1}|,...,|c_{n}|)^{-\nu }} {\displaystyle 0<|c_{1}x_{1}+...+c_{n}x_{n}|\leq \max(|c_{1}|,...,|c_{n}|)^{-\nu }} with c 1 , . . . , c n ∈ Z {\displaystyle c_{1},...,c_{n}\in \mathbb {Z} } {\displaystyle c_{1},...,c_{n}\in \mathbb {Z} } and ν ∈ R 0 + {\displaystyle \nu \in \mathbb {R} _{0}^{+}} {\displaystyle \nu \in \mathbb {R} _{0}^{+}}.

Define R {\displaystyle R} {\displaystyle R} as the set of all ν ∈ R 0 + {\displaystyle \nu \in \mathbb {R} _{0}^{+}} {\displaystyle \nu \in \mathbb {R} _{0}^{+}} for which infinitely many solutions c 1 , . . . c n ∈ Z {\displaystyle c_{1},...c_{n}\in \mathbb {Z} } {\displaystyle c_{1},...c_{n}\in \mathbb {Z} } exist, such that the inequality is satisfied. Then ν ( x 1 , . . . , x n ) = sup R {\displaystyle \nu (x_{1},...,x_{n})=\sup R} {\displaystyle \nu (x_{1},...,x_{n})=\sup R} is the linear independence measure.

If the x 1 , . . . , x n {\displaystyle x_{1},...,x_{n}} {\displaystyle x_{1},...,x_{n}} are linearly dependent over Q {\displaystyle \mathbb {\mathbb {Q} } } {\displaystyle \mathbb {\mathbb {Q} } } then ν ( x 1 , . . . , x n ) = 0 {\displaystyle \nu (x_{1},...,x_{n})=0} {\displaystyle \nu (x_{1},...,x_{n})=0}.

If 1 , x 1 , . . . , x n {\displaystyle 1,x_{1},...,x_{n}} {\displaystyle 1,x_{1},...,x_{n}} are linearly independent algebraic numbers over Q {\displaystyle \mathbb {\mathbb {Q} } } {\displaystyle \mathbb {\mathbb {Q} } } then ν ( 1 , x 1 , . . . , x n ) ≤ n {\displaystyle \nu (1,x_{1},...,x_{n})\leq n} {\displaystyle \nu (1,x_{1},...,x_{n})\leq n}.[33]

It is further ν ( 1 , x ) = ω 1 ( x ) = μ ( x ) − 1 {\displaystyle \nu (1,x)=\omega _{1}(x)=\mu (x)-1} {\displaystyle \nu (1,x)=\omega _{1}(x)=\mu (x)-1}.

Other generalizations

Koksma’s generalization

Jurjen Koksma in 1939 proposed another generalization, similar to that of Mahler, based on approximations of complex numbers by algebraic numbers.[3][38]

For a given complex number z {\displaystyle z} {\displaystyle z} consider algebraic numbers α {\displaystyle \alpha } {\displaystyle \alpha } of degree at most n {\displaystyle n} {\displaystyle n}. Define a height function H ( α ) = H ( P ) {\displaystyle H(\alpha )=H(P)} {\displaystyle H(\alpha )=H(P)}, where P {\displaystyle P} {\displaystyle P} is the characteristic polynomial of α {\displaystyle \alpha } {\displaystyle \alpha } and consider the inequality:

0 < | z − α | ≤ H ( α ) − ω ∗ − 1 {\displaystyle 0<|z-\alpha |\leq H(\alpha )^{-\omega ^{*}-1}} {\displaystyle 0<|z-\alpha |\leq H(\alpha )^{-\omega ^{*}-1}} with ω ∗ ∈ R 0 + {\displaystyle \omega ^{*}\in \mathbb {R} _{0}^{+}} {\displaystyle \omega ^{*}\in \mathbb {R} _{0}^{+}}.

Set R {\displaystyle R} {\displaystyle R} to be the set of all ω ∗ ∈ R 0 + {\displaystyle \omega ^{*}\in \mathbb {R} _{0}^{+}} {\displaystyle \omega ^{*}\in \mathbb {R} _{0}^{+}} for which infinitely many such algebraic numbers α {\displaystyle \alpha } {\displaystyle \alpha } exist, that keep the inequality satisfied. Further define ω n ∗ ( z ) = sup R {\displaystyle \omega _{n}^{*}(z)=\sup R} {\displaystyle \omega _{n}^{*}(z)=\sup R} for all n ∈ Z + {\displaystyle n\in \mathbb {Z} ^{+}} {\displaystyle n\in \mathbb {Z} ^{+}} with ω 1 ∗ ( z ) {\displaystyle \omega _{1}^{*}(z)} {\displaystyle \omega _{1}^{*}(z)} being an irrationality measure, ω 2 ∗ ( z ) {\displaystyle \omega _{2}^{*}(z)} {\displaystyle \omega _{2}^{*}(z)} being a non-quadraticity measure,[17] etc.

Then Koksma's transcendence measure is given by:

ω ∗ ( z ) = lim sup n → ∞ ω n ∗ ( z ) {\displaystyle \omega ^{*}(z)=\limsup _{n\to \infty }\omega _{n}^{*}(z)} {\displaystyle \omega ^{*}(z)=\limsup _{n\to \infty }\omega _{n}^{*}(z)}.

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.[38]: 87 

Simultaneous approximation of real numbers

Given a real number x ∈ R {\displaystyle x\in \mathbb {R} } {\displaystyle x\in \mathbb {R} }, an irrationality measure of x {\displaystyle x} {\displaystyle x} quantifies how well it can be approximated by rational numbers p q {\displaystyle {\frac {p}{q}}} {\displaystyle {\frac {p}{q}}} with denominator q ∈ Z + {\displaystyle q\in \mathbb {Z} ^{+}} {\displaystyle q\in \mathbb {Z} ^{+}}. If x = α {\displaystyle x=\alpha } {\displaystyle x=\alpha } is taken to be an algebraic number that is also irrational one may obtain that the inequality

0 < | α − p q | < 1 q μ {\displaystyle 0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}} {\displaystyle 0<\left|\alpha -{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}}}

has only at most finitely many solutions p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } for μ > 2 {\displaystyle \mu >2} {\displaystyle \mu >2}. This is known as Roth's theorem.

This can be generalized: Given a set of real numbers x 1 , . . . , x n ∈ R {\displaystyle x_{1},...,x_{n}\in \mathbb {R} } {\displaystyle x_{1},...,x_{n}\in \mathbb {R} } one can quantify how well they can be approximated simultaneously by rational numbers p 1 q , . . . , p n q {\displaystyle {\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}} {\displaystyle {\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}} with the same denominator q ∈ Z + {\displaystyle q\in \mathbb {Z} ^{+}} {\displaystyle q\in \mathbb {Z} ^{+}}. If the x i = α i {\displaystyle x_{i}=\alpha _{i}} {\displaystyle x_{i}=\alpha _{i}} are taken to be algebraic numbers, such that 1 , α 1 , . . . , α n {\displaystyle 1,\alpha _{1},...,\alpha _{n}} {\displaystyle 1,\alpha _{1},...,\alpha _{n}} are linearly independent over the rational numbers Q {\displaystyle \mathbb {Q} } {\displaystyle \mathbb {Q} } it follows that the inequalities

0 < | α i − p i q | < 1 q μ , ∀ i ∈ { 1 , . . . , n } {\displaystyle 0<\left|\alpha _{i}-{\frac {p_{i}}{q}}\right|<{\frac {1}{q^{\mu }}},\forall i\in \{1,...,n\}} {\displaystyle 0<\left|\alpha _{i}-{\frac {p_{i}}{q}}\right|<{\frac {1}{q^{\mu }}},\forall i\in \{1,...,n\}}

have only at most finitely many solutions ( p 1 q , . . . , p n q ) ∈ Q n {\displaystyle \left({\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}\right)\in \mathbb {Q} ^{n}} {\displaystyle \left({\frac {p_{1}}{q}},...,{\frac {p_{n}}{q}}\right)\in \mathbb {Q} ^{n}} for μ > 1 + 1 n {\displaystyle \mu >1+{\frac {1}{n}}} {\displaystyle \mu >1+{\frac {1}{n}}}. This result is due to Wolfgang M. Schmidt.[40][41]

See also

References

  1. Sondow, Jonathan (2004). "Irrationality Measures, Irrationality Bases, and a Theorem of Jarnik". arXiv:math/0406300.
  2. Parshin, A. N.; Shafarevich, I. R. (2013-03-09). Number Theory IV: Transcendental Numbers. Springer Science & Business Media. ISBN 978-3-662-03644-0.
  3. Bugeaud, Yann (2012). Distribution modulo one and Diophantine approximation. Cambridge Tracts in Mathematics. Vol. 193. Cambridge: Cambridge University Press. doi:10.1017/CBO9781139017732. ISBN 978-0-521-11169-0. MR 2953186. Zbl 1260.11001.
  4. Tao, Terence (2009). "245B, Notes 9: The Baire category theorem and its Banach space consequences". What's new. Retrieved 2024-09-08.
  5. Borwein, Jonathan M. (1987). Pi and the AGM: A Study in Analytic Number Theory and Computational Complexity. Wiley.
  6. Sondow, Jonathan (2003-07-23). "An irrationality measure for Liouville numbers and conditional measures for Euler's constant". arXiv:math/0307308.
  7. Chudnovsky, G. V. (1982). "Hermite-padé approximations to exponential functions and elementary estimates of the measure of irrationality of π". In Chudnovsky, David V.; Chudnovsky, Gregory V. (eds.). The Riemann Problem, Complete Integrability and Arithmetic Applications. Lecture Notes in Mathematics. Vol. 925. Berlin, Heidelberg: Springer. pp. 299–322. doi:10.1007/BFb0093516. ISBN 978-3-540-39152-4.
  8. Shallit, Jeffrey (1979-05-01). "Simple continued fractions for some irrational numbers". Journal of Number Theory. 11 (2): 209–217. doi:10.1016/0022-314X(79)90040-4. ISSN 0022-314X.
  9. Shallit, J. O (1982-04-01). "Simple continued fractions for some irrational numbers, II". Journal of Number Theory. 14 (2): 228–231. doi:10.1016/0022-314X(82)90047-6. ISSN 0022-314X.
  10. Bugeaud, Yann (2011). "On the rational approximation to the Thue–Morse–Mahler numbers". Annales de l'Institut Fourier. 61 (5): 2065–2076. doi:10.5802/aif.2666. ISSN 1777-5310.
  11. Weisstein, Eric W. "Irrationality Measure". mathworld.wolfram.com. Retrieved 2020-10-14.
  12. Nesterenko, Yu. V. (2010-10-01). "On the irrationality exponent of the number ln 2". Mathematical Notes. 88 (3): 530–543. doi:10.1134/S0001434610090257. ISSN 1573-8876. S2CID 120685006.
  13. Wu, Qiang (2003). "On the Linear Independence Measure of Logarithms of Rational Numbers". Mathematics of Computation. 72 (242): 901–911. doi:10.1090/S0025-5718-02-01442-4. ISSN 0025-5718. JSTOR 4099938.
  14. Bouchelaghem, Abderraouf; He, Yuxin; Li, Yuanhang; Wu, Qiang (2024-03-01). "On the linear independence measures of logarithms of rational numbers. II". J. Korean Math. Soc. 61 (2): 293–307. doi:10.4134/JKMS.j230133.
  15. Sal’nikova, E. S. (2008-04-01). "Diophantine approximations of log 2 and other logarithms". Mathematical Notes. 83 (3): 389–398. doi:10.1134/S0001434608030097. ISSN 1573-8876.
  16. "Symmetrized polynomials in a problem of estimating of the irrationality measure of number ln 3". www.mathnet.ru. Retrieved 2020-10-14.
  17. Polyanskii, Alexandr (2015-01-27). "On the irrationality measure of certain numbers". arXiv:1501.06752 [math.NT].
  18. Polyanskii, A. A. (2018-03-01). "On the Irrationality Measures of Certain Numbers. II". Mathematical Notes. 103 (3): 626–634. doi:10.1134/S0001434618030306. ISSN 1573-8876. S2CID 125251520.
  19. Androsenko, V. A. (2015). "Irrationality measure of the number \frac{\pi}{\sqrt{3}}". Izvestiya: Mathematics. 79 (1): 1–17. doi:10.1070/im2015v079n01abeh002731. ISSN 1064-5632. S2CID 123775303.
  20. Zeilberger, Doron; Zudilin, Wadim (2020-01-07). "The irrationality measure of π is at most 7.103205334137...". Moscow Journal of Combinatorics and Number Theory. 9 (4): 407–419. arXiv:1912.06345. doi:10.2140/moscow.2020.9.407. S2CID 209370638.
  21. Alekseyev, Max A. (2011). "On convergence of the Flint Hills series". arXiv:1104.5100 [math.CA].
  22. Weisstein, Eric W. "Flint Hills Series". MathWorld.
  23. Meiburg, Alex (2022). "Bounds on Irrationality Measures and the Flint-Hills Series". arXiv:2208.13356 [math.NT].
  24. OpenAI (2026-09-24). "The irrationality exponent of π is 2" (PDF). GitHub. Retrieved 2026-10-07.
  25. Zudilin, Wadim (2014-06-01). "Two hypergeometric tales and a new irrationality measure of ζ(2)". Annales mathématiques du Québec. 38 (1): 101–117. arXiv:1310.1526. doi:10.1007/s40316-014-0016-0. ISSN 2195-4763. S2CID 119154009.
  26. Bashmakova, M. G.; Salikhov, V. Kh. (2019). "Об оценке меры иррациональности arctg 1/2". Чебышевский сборник. 20 (4 (72)): 58–68. ISSN 2226-8383.
  27. Tomashevskaya, E. B. "On the irrationality measure of the number log 5+pi/2 and some other numbers". www.mathnet.ru. Retrieved 2020-10-14.
  28. Salikhov, Vladislav K.; Bashmakova, Mariya G. (2022). "On rational approximations for some values of arctan(s/r) for natural s and r, s". Moscow Journal of Combinatorics and Number Theory. 11 (2): 181–188. doi:10.2140/moscow.2022.11.181. ISSN 2220-5438.
  29. Salikhov, V. Kh.; Bashmakova, M. G. (2020-12-01). "On Irrationality Measure of Some Values of $\operatorname{arctg} \frac{1}{n}$". Russian Mathematics. 64 (12): 29–37. doi:10.3103/S1066369X2012004X. ISSN 1934-810X.
  30. Bruiltet, Sylvain (2002). "D'une mesure d'approximation simultanee a une mesure d'irrationalite : le cas de Γ(1/4) et Γ(1/3)". Acta Arithmetica. 104 (3): 243–281. doi:10.4064/aa104-3-3. ISSN 0065-1036.
  31. Duverney, Daniel; Shiokawa, Iekata (2020-01-01). "Irrationality exponents of numbers related with Cahen's constant". Monatshefte für Mathematik. 191 (1): 53–76. doi:10.1007/s00605-019-01335-0. ISSN 1436-5081.
  32. Amou, Masaaki (1991-02-01). "Approximation to certain transcendental decimal fractions by algebraic numbers". Journal of Number Theory. 37 (2): 231–241. doi:10.1016/S0022-314X(05)80039-3. ISSN 0022-314X.
  33. Waldschmidt, Michel (2004-01-24). "Open Diophantine Problems". arXiv:math/0312440.
  34. Hurwitz, A. (1891). "Ueber die angenäherte Darstellung der Irrationalzahlen durch rationale Brüche (On the approximate representation of irrational numbers by rational fractions)". Mathematische Annalen (in German). 39 (2): 279–284. doi:10.1007/BF01206656. JFM 23.0222.02. S2CID 119535189.
  35. LeVeque, William (1977). Fundamentals of Number Theory. Addison-Wesley Publishing Company, Inc. pp. 251–254. ISBN 0-201-04287-8.
  36. Hancl, Jaroslav (January 2016). "Second basic theorem of Hurwitz". Lithuanian Mathematical Journal. 56: 72–76. doi:10.1007/s10986-016-9305-4. S2CID 124639896.
  37. Davis, C. S. (1978). "Rational approximations to e". Journal of the Australian Mathematical Society. 25 (4): 497–502. doi:10.1017/S1446788700021480. ISSN 1446-8107.
  38. Baker, Alan (1979). Transcendental number theory (Repr. with additional material ed.). Cambridge: Cambridge Univ. Pr. ISBN 978-0-521-20461-3.
  39. Burger, Edward B.; Tubbs, Robert (2004-07-28). Making Transcendence Transparent: An Intuitive Approach to Classical Transcendental Number Theory. Springer Science & Business Media. ISBN 978-0-387-21444-3.
  40. Schmidt, Wolfgang M. (1972). "Norm Form Equations". Annals of Mathematics. 96 (3): 526–551. doi:10.2307/1970824. ISSN 0003-486X. JSTOR 1970824.
  41. Schmidt, Wolfgang M. (1996). Diophantine approximation. Lecture notes in mathematics. Berlin; New York: Springer. ISBN 978-3-540-09762-4.