Lochs' theorem

In number theory, the set of hole is a theorem on the convergence rate of the continued fraction representations of real numbers. The theorem was proved in 1964 by Gustav hole. Then the continued fraction notation is only slightly more efficient than the decimal representation.

The set

The theorem states that for almost all real numbers in the interval showing the number of terms of the continued fraction representation of a number, which is required to the first digits of the decimal representation of the number, behaves asymptototisch as follows:

(Fractions 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 each new term in the continued fraction representation of a "normal" real number (good) increases the Right of the representation by about one decimal place. For the circle number about 968 partial denominators of the continued fraction expansion lead to a precision of 1000 decimal places ( cf. Pi continued fraction representation).

In other places systems

The decimal system is the last place value system, in which a new point less " value " than bringing a new quotient of the continued fraction representation; the penalty system (replace by in the formula), the value is slightly greater than 1:

More

The reciprocal value of the limiting value for the decimal system, that

Is twice the logarithm of the Lévyschen constant.

Credentials

• Eric W. Weisstein: hole ' theorem. In: MathWorld (English).
• Karma Dajani and Cor Kraaikamp: Ergodic theory of numbers, Cambridge University Press, 2002, ISBN 0883850346, 9780883850343, Incomplete online version ( Google books)
• C. Faivre: A central limit theorem related to decimal and continued fraction expansion in Arch Math 70 (1998) pp. 455-463 ( springerlink.com )

• Analysis
• Set ( mathematics)