A Central Limit Theorem Related to Continued Fraction Expansions of Two Close Real Numbers

Abstract

This thesis generalizes the central limit theorem initially proposed by C. Faivre, which pertains to continued fraction expansions. For a given real number x in the interval [0,1), the quantity kn(x) represents the maximum number of consecutive matching digits in the continued fraction expansions of two numbers y and z, where y and z are the lower and upper decimal approximations of x at the n-th decimal place, respectively. Faivre demonstrated that the appropriately normalized distribution of kn(x) converges to the standard normal distribution as n goes to infinity.

Building on this, the study introduces kε(x), defined as the maximum number of matching digits in the continued fraction expansions of x and x + ε , where ε > 0 and ε → 0. The thesis establishes that the normalized distribution of kε(x) adheres to the central limit theorem, thereby generalizing Faivre’s results. Crucially, this work emphasizes that the analysis does not depend on comparing different ex- pansion systems; instead, the focus lies on the asymptotic behavior of two points x and x + ε as they converge arbitrarily close to each other.