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Lochs' Theorem


Lochs' theorem states that for almost every irrational number x in (0,1) with respect to Lebesgue measure, the number m=m(n,x) of initial partial denominators of its regular continued fraction determined by the first n digits of its decimal expansion satisfies

lim_(n->infty)m/n=(6ln2ln10)/(pi^2)
(1)
=0.970270114392...
(2)

(OEIS A086819; Lochs 1964). This number is sometimes known as Lochs' constant.

Thus for large n, the first n digits of a decimal expansion determine approximately 0.97027n initial partial denominators for almost every irrational number.


See also

Khinchin's Constant, Lochs' Constant, Regular Continued Fraction

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References

Bosma, W.; Dajani, K.; and Kraaikamp, C. "Entropy and Counting Correct Digits." Univ. Nijmegen Math. Report 9925, 1999.Finch, S. R. Mathematical Constants. Cambridge, England: Cambridge University Press, 2003.Kintchine, A. "Zur metrischen Kettenbruchtheorie." Compos. Math. 3, 276-285, 1936.Kraaikamp, C. "A New Class of Continued Fraction Expansions." Acta Arith. 57, 1-39, 1991.Lévy, P. "Sur le developpement en fraction continue d'un nombre choisi au hasard." Compos. Math. 3, 286-303, 1936.Lochs, G. "Vergleich der Genauigkeit von Dezimalbruch und Kettenbruch." Abh. Hamburg Univ. Math. Sem. 27, 142-144, 1964.Perron, O. Die Lehre von den Kettenbrüchen, 3. verb. und erweiterte Aufl. Stuttgart, Germany: Teubner, 1954-57.Sloane, N. J. A. Sequence A086819 in "The On-Line Encyclopedia of Integer Sequences."

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Lochs' Theorem

Cite this as:

Weisstein, Eric W. "Lochs' Theorem." From MathWorld--A Wolfram Resource. https://mathworld.wolfram.com/LochsTheorem.html

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