In mathematics, an irrationality measure of a real number x {\displaystyle x} is a measure of how "closely" it can be approximated by rationals. If a function f ( t , λ ) {\displaystyle f(t,\lambda )} , defined for t , λ > 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 )}
for a given real number x ∈ R {\displaystyle x\in \mathbb {R} } and rational numbers p q {\displaystyle {\frac {p}{q}}} with p ∈ Z , q ∈ Z + {\displaystyle p\in \mathbb {Z} ,q\in \mathbb {Z} ^{+}} . For x ∈ R {\displaystyle x\in \mathbb {R} } , define R {\displaystyle R} as the set of all λ ∈ 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\}} is finite. Then λ ( x ) = inf R {\displaystyle \lambda (x)=\inf R} is called an irrationality measure of x {\displaystyle x} with regard to f . {\displaystyle f.} If the set R {\displaystyle R} is empty, x {\displaystyle x} is said to have infinite irrationality measure λ ( x ) = ∞ {\displaystyle \lambda (x)=\infty } . By definition, there is a sequence ( ε n ) n ∈ N {\displaystyle (\varepsilon _{n})_{n\in \mathbb {N} }} converging to 0 such that for every 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})}
has at most finitely many solutions p q {\displaystyle {\frac {p}{q}}} . By the nondecreasing property of 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 )}
has at most only finitely many solutions p q {\displaystyle {\frac {p}{q}}} for all ε > 0 {\displaystyle \varepsilon >0} .
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 }} , a definition adapting the one of Liouville numbers — the irrationality exponent μ ( x ) {\displaystyle \mu (x)} is defined for real numbers x {\displaystyle x} to be the supremum of the set of μ {\displaystyle \mu } such that 0 < | x − p q | < 1 q μ {\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)} with q > 0 {\displaystyle q>0} . For any value n < μ ( x ) {\displaystyle n<\mu (x)} , the infinite set of all rationals p / q {\displaystyle p/q} satisfying the above inequality yields good approximations of x {\displaystyle x} . Conversely, if n > μ ( x ) {\displaystyle n>\mu (x)} , then there are at most finitely many coprime ( p , q ) {\displaystyle (p,q)} with q > 0 {\displaystyle q>0} that satisfy the inequality. For example, whenever a rational approximation p q ≈ x {\displaystyle {\frac {p}{q}}\approx x} with p , q ∈ N {\displaystyle p,q\in \mathbb {N} } yields 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 }}}}
for any ε > 0 {\displaystyle \varepsilon >0} , except for at most a finite number of "lucky" pairs ( p , q ) {\displaystyle (p,q)} . A number x ∈ R {\displaystyle x\in \mathbb {R} } with irrationality exponent μ ( x ) ≤ 2 {\displaystyle \mu (x)\leq 2} is called a diophantine number, while numbers with μ ( x ) = ∞ {\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. It is μ ( x ) = μ ( r x + s ) {\displaystyle \mu (x)=\mu (rx+s)} for real numbers x {\displaystyle x} and rational numbers r ≠ 0 {\displaystyle r\neq 0} and s {\displaystyle s} . If for some x {\displaystyle x} we have μ ( x ) ≤ μ {\displaystyle \mu (x)\leq \mu } , then it follows μ ( x 1 / 2 ) ≤ 2 μ {\displaystyle \mu (x^{1/2})\leq 2\mu } . For a real number 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},...]} with convergents p i / q i {\displaystyle p_{i}/q_{i}} it holds:
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}}}.}
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 } 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 } for some positive real numbers σ , τ {\displaystyle \sigma ,\tau } , then we can establish an upper bound for the irrationality exponent of x {\displaystyle x} by:
μ ( x ) ≤ 1 + σ τ {\displaystyle \mu (x)\leq 1+{\frac {\sigma }{\tau }}}
Known bounds For most transcendental numbers, the exact value of their irrationality exponent is not known. Below is a table of known upper and lower bounds.
Irrationality base The irrationality base or Sondow irrationality measure is obtained by setting f ( q , β ) = β − q {\displaystyle f(q,\beta )=\beta ^{-q}} . 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} for all other real numbers: Let x {\displaystyle x} be an irrational number. If there exist real numbers β ≥ 1 {\displaystyle \beta \geq 1} with the property that for any ε > 0 {\displaystyle \varepsilon >0} , there is a positive integer q ( ε ) {\displaystyle q(\varepsilon )} such that
| x − p q | > 1 ( β + ε ) q {\displaystyle \left|x-{\frac {p}{q}}\right|>{\frac {1}{(\beta +\varepsilon )^{q}}}}
for all integers p , q {\displaystyle p,q} with q ≥ q ( ε ) {\displaystyle q\geq q(\varepsilon )} then the least such β {\displaystyle \beta } is called the irrationality base of x {\displaystyle x} and is represented as β ( x ) {\displaystyle \beta (x)} . If no such β {\displaystyle \beta } exists, then β ( x ) = ∞ {\displaystyle \beta (x)=\infty } and x {\displaystyle x} is called a super Liouville number. If a real number 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},...]} with convergents 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}}}} .
Examples Any real number x {\displaystyle x} with finite irrationality exponent μ ( x ) < ∞ {\displaystyle \mu (x)<\infty } has irrationality base β ( x ) = 1 {\displaystyle \beta (x)=1} , while any number with irrationality base β ( x ) > 1 {\displaystyle \beta (x)>1} has irrationality exponent μ ( x ) = ∞ {\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}},...]} has irrationality exponent μ ( L ) = ∞ {\displaystyle \mu (L)=\infty } and irrationality base β ( 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}}}}}+...} ( n a {\displaystyle {^{n}a}} represents tetration, a = 2 , 3 , 4... {\displaystyle a=2,3,4...} ) have irrationality base β ( τ 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 } has irrationality base β ( S ) = ∞ {\displaystyle \beta (S)=\infty } , hence it is a super Liouville number. Although it is not known whether or not e π {\displaystyle e^{\pi }} is a Liouville number, it is known that β ( e π ) = 1 {\displaystyle \beta (e^{\pi })=1} .
Other irrationality measures
Markov constant
Setting f ( q , M ) = ( M q 2 ) − 1 {\displaystyle f(q,M)=(Mq^{2})^{-1}} gives a stronger irrationality measure: the Markov constant M ( x ) {\displaystyle M(x)} . For an irrational number x ∈ R ∖ 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} . Namely if 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}}}}
has infinitely many solutions p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } . If 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} and Hurwitz's theorem gives M ( x ) ≥ 5 {\displaystyle M(x)\geq {\sqrt {5}}} both for irrational x {\displaystyle x} . This is in fact the best general lower bound since the golden ratio gives M ( φ ) = 5 {\displaystyle M(\varphi )={\sqrt {5}}} . It is also M ( 2 ) = 2 2 {\displaystyle M({\sqrt {2}})=2{\sqrt {2}}} . Given x = [ a 0 ; a 1 , a 2 , . . . ] {\displaystyle x=[a_{0};a_{1},a_{2},...]} by its simple continued fraction expansion, one may obtain:
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}])}.}
Bounds for the Markov constant of 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} with p = lim sup n → ∞ a n {\displaystyle p=\limsup _{n\to \infty }a_{n}} . This implies that M ( x ) = ∞ {\displaystyle M(x)=\infty } if and only if ( a k ) {\displaystyle (a_{k})} is not bounded. The numbers with finite M ( x ) {\displaystyle M(x)} are called badly approximable. In particular every quadratic irrational number x {\displaystyle x} is badly approximable. Any number with μ ( x ) > 2 {\displaystyle \mu (x)>2} or β ( x ) > 1 {\displaystyle \beta (x)>1} has an unbounded simple continued fraction and hence M ( x ) = ∞ {\displaystyle M(x)=\infty } . A further consequence is M ( e ) = ∞ {\displaystyle M(e)=\infty } . For rational numbers r {\displaystyle r} it may be defined M ( r ) = 0 {\displaystyle M(r)=0} .
Other results The values M ( e ) = ∞ {\displaystyle M(e)=\infty } and μ ( 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}}}} has for all c ∈ R + {\displaystyle c\in \mathbb {R} ^{+}} infinitely many solutions p q ∈ 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 }}}} has for all ε ∈ R + {\displaystyle \varepsilon \in \mathbb {R} ^{+}} only at most finitely many solutions p q ∈ 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:
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}}}
which is satisfied by infinitely many p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } for c > 1 2 {\displaystyle c>{\tfrac {1}{2}}} but not for c < 1 2 {\displaystyle c<{\tfrac {1}{2}}} . This makes the number 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} } the inequality below has infinitely many solutions p q ∈ Q {\displaystyle {\frac {p}{q}}\in \mathbb {Q} } : (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}}}
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.
Mahler's irrationality measure Instead of taking for a given real number x {\displaystyle x} the difference | x − p / q | {\displaystyle |x-p/q|} with p / q ∈ Q {\displaystyle p/q\in \mathbb {Q} } , one may instead focus on term | q x − p | = | L ( x ) | {\displaystyle |qx-p|=|L(x)|} with p , q ∈ Z {\displaystyle p,q\in \mathbb {Z} } and L ∈ Z [ x ] {\displaystyle L\in \mathbb {Z} [x]} with 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 }} with p , q ∈ Z {\displaystyle p,q\in \mathbb {Z} } and ω ∈ R 0 + {\displaystyle \omega \in \mathbb {R} _{0}^{+}} . Define R {\displaystyle R} as the set of all ω ∈ R 0 + {\displaystyle \omega \in \mathbb {R} _{0}^{+}} for which infinitely many solutions