Lochs' theorem

In number theory, Lochs' theorem is a theorem concerning the rate of convergence of the continued fraction expansion of a typical real number. A proof of the theorem was published by Gustav Lochs in 1964.^[1]

The theorem states that for almost all real numbers in the interval (0,1), the number of terms m of the number's continued fraction expansion that are required to determine the first n places of the number's decimal expansion behaves asymptotically as follows:

${\displaystyle \lim _{n\to \infty }{\frac {m}{n}}={\frac {6\ln(2)\ln(10)}{\pi ^{2}}}\approx 0.97027014}$ (sequence A086819 in the OEIS).^[2]

As this limit is only slightly smaller than 1, this can be interpreted as saying that each additional term in the continued fraction representation of a "typical" real number increases the accuracy of the representation by approximately one decimal place. The decimal system is the last positional system for which each digit carries less information than one continued fraction quotient; going to base-11 (changing ${\displaystyle \ln(10)}$ to ${\displaystyle \ln(11)}$ in the equation) makes the above value exceed 1.

The reciprocal of this limit,

${\displaystyle {\frac {\pi ^{2}}{6\ln(2)\ln(10)}}\approx 1.03064083}$ (sequence A062542 in the OEIS),

is twice the base-10 logarithm of Lévy's constant.

Three typical numbers, and the golden ratio. The typical numbers follow an approximately 45° line, since each continued fraction coefficient yields approximately one decimal digit. The golden ratio, on the other hand, is the number requiring the most coefficients for each digit.

A prominent example of a number not exhibiting this behavior is the golden ratio—sometimes known as the "most irrational" number—whose continued fraction terms are all ones, the smallest possible in canonical form. On average it requires approximately 2.39 continued fraction terms per decimal digit.^[3]

References

1. Lochs, Gustav (1964), "Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch", Abhandlungen aus dem Mathematischen Seminar der Universität Hamburg (in German), 27: 142–144, doi:10.1007/BF02993063, MR 0162753.
2. Weisstein, Eric W. "Lochs' Theorem". MathWorld.
3. Cooper, Harold. "Continued Fraction Streams". Retrieved 30 August 2016.