Questions tagged [irrationality-measure]

The Liouville-Roth irrationality measure (irrationality exponent, approximation exponent, or Liouville–Roth constant) of a real number $x$ is a measure of how "closely" it can be approximated by rationals. The largest possible value for $\mu$ such that $ 0<\left|x-{\frac {p}{q}}\right|<{\frac {1}{q^{\mu }}} $ is satisfied by an infinite number of integer pairs $(p, q)$ with $q > 0$ is defined to be the irrationality measure of $x$. For any value $\mu$ less than this upper bound, the infinite set of all rationals $\frac p q$ satisfying the above inequality yield an approximation of $x$. Conversely, if $\mu$ is greater than the upper bound, then there are at most finitely many $(p, q)$ with $q > 0$ that satisfy the inequality; thus, the opposite inequality holds for all larger values of $q$. In other words, given the irrationality measure $\mu$ of a real number $x$, whenever a rational approximation $x \approx \frac p q$, $p,q \in \mathbb Z$ and $q>0$, yields $n + 1$ exact decimal digits, we have

$$ \frac {1}{10^{n}}\ge \left|x-\frac {p}{q}\right|\ge \frac {1}{q^{\mu +\epsilon }} $$

for any $\epsilon>0$, except for at most a finite number of "lucky" pairs $(p, q)$.

For a rational number $\alpha$ the irrationality measure is $\mu(\alpha) = 1$. The Thue–Siegel–Roth theorem states that if $\alpha$ is an algebraic number, real but not rational, then $\mu(\alpha) = 2$.

Almost all numbers have an irrationality measure equal to $2$.

Transcendental numbers have irrationality measure $2$ or greater. For example, the transcendental number $e$ has $\mu(e) = 2$. The irrationality measures of $\pi$, $\log 2$, and $\log 3$ are at most $7.103205334137$, $3.57455391$, and $5.125$, respectively.

It has been proven that if the series $ \sum _{n=1}^{\infty }{\frac {\csc ^{2}n}{n^{3}}} $ (where $n$ is in radians) converges, then $ \pi $'s irrationality measure is at most $2.5$.

The Liouville numbers are precisely those numbers having infinite irrationality measure.

Source: Wikipedia