Lochs's theorem

Lochs's theorem states that, for almost every real number, its first n decimal digits determine about (6 ln 2 ln 10/π²)n terms of its continued fraction as n grows.

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Continued fraction: A representation of a number as an integer plus a sequence of reciprocals of integers. Its initial terms are the quantities whose growth Lochs's theorem estimates.
Continued-fraction convergent: A rational approximation obtained by truncating a number's continued fraction. Convergents mark the successive precision levels represented by continued-fraction prefixes.
Alfréd Rényi: A Hungarian mathematician who made foundational contributions to probability theory and information theory. Rényi developed an early information-theoretic approach to the relation between number representations.
Continued-fraction expansion: The sequence of partial quotients representing a real number through repeated reciprocation and integer extraction. The theorem gives a precise conversion rate from decimal digits to this expansion's initial terms.
Cylinder set: A set of points sharing a specified finite initial segment under a symbolic or dynamical coding. Fixing decimal digits creates an interval whose continued-fraction coding determines how many terms are known.
Almost everywhere: A property holds almost everywhere when its exceptions form a set of measure zero. This quantifier permits exceptional numbers without changing the theorem's typical asymptotic claim.
Wacław Sierpiński: A Polish mathematician known for major work in set theory, topology, and number theory. Sierpiński's work on decimal and continued-fraction expansions forms part of the theorem's historical background.
Diophantine approximation: The study of approximating real numbers by rational numbers and the quality of those approximations. Continued-fraction prefixes encode strong rational approximations, linking the theorem to approximation precision.
Gauss map: The map x ↦ 1/x − ⌊1/x⌋ on (0,1), which shifts the continued-fraction digits of a number. Its invariant statistics describe the typical continued-fraction terms counted by the theorem.
Lebesgue measure: The standard notion of length, area, and volume extended to measurable sets. It specifies the sense in which the theorem's exceptional real numbers have measure zero.