Set of holes

In number theory , Loch's theorem is a theorem about the rate of convergence of continued fraction representations of real numbers . The theorem was proved by Gustav Lochs in 1964 . According to this, the continued fraction notation is only slightly more efficient than the representation of decimal numbers.

The sentence

The theorem states that for almost all real numbers in the interval, the number of terms in the continued fraction representation of a number, which is required to represent the first digits of the decimal representation of the number, behaves asymptotically as follows: ${\displaystyle (0,1)}$ ${\displaystyle m}$ ${\displaystyle n}$

${\displaystyle \lim _{n\to \infty }{\frac {m}{n}}={\frac {6\cdot \ln 2\cdot \ln 10}{\pi ^{2}}}=0{,}970270114...}$

( Decimal places of the value: sequence A086819 in OEIS )

The set of numbers for which this does not apply has the Lebesgue measure zero .

Since this limit is only slightly smaller than 1, it can be said that every new term in the continued fraction representation of a "normal" real number increases the accuracy of the representation by about (good) one decimal place. For the circle number, for example, 968 partial denominators of the continued fraction expansion lead to an accuracy of 1000 decimal places (see Pi continued fraction representation ). ${\displaystyle \pi }$

In other place value systems

The decimal system is the last place value system in which a new digit brings less "value" than a new quotient of the continued fraction representation; In the penalty system (replace with in the formula) the value is slightly larger than 1: ${\displaystyle \ln 10}$ ${\displaystyle \ln 11}$

Basis of the ranking system	Limit value (a new place in the place value system corresponds on average to ... part denominators in the continued fraction representation)
2	0.2920804083 ...
3	0.4629364943 ...
4th	0.5841608166 ...
...
10	0.9702701143 ...
11	1.0104321997…
12	1.0470973110 ...
13	1.0808259438 ...
...
20th	1.2623505227 ...
100	1.9405402287 ...
${\displaystyle \to \infty }$	${\displaystyle \to \infty }$

additional

The reciprocal of the limit for the decimal system, that is

${\displaystyle {\frac {\pi ^{2}}{6\cdot \ln 2\cdot \ln 10}}=1{,}03064083...}$,

is twice the logarithm of ten of Lévy's constant .

literature

• Karma Dajani, Cor Kraaikamp: Ergodic theory of numbers. Cambridge University Press, 2002, ISBN 0-88385-034-6 , books.google.de
• C. Faivre: A central limit theorem related to decimal and continued fraction expansion. In: Arch. Math. 70, 1998, pp. 455-463, springerlink.com

Web links

• Eric W. Weisstein : Lochs' Theorem . In: MathWorld (English).

Individual evidence

1. ↑ G. Lochs: Abh. Hamburg Univ. Math. Sem. 27, 1964, pp. 142-144.
2. ↑ Follow A062542 in OEIS
3. ↑ lacim.uqam.ca ( Memento from March 17, 2011 in the Internet Archive )