Beta-expansion and continued fraction expansion of real numbers

TL;DR

This paper investigates the relationship between beta-expansions and continued fraction expansions of real numbers, showing exponential decay in deviation probabilities and comparing their approximation qualities.

Contribution

It extends previous results by proving exponential decay of deviation measures for all beta > 1 and compares the approximation effectiveness of beta-expansions versus continued fractions.

Findings

Deviation probability decays exponentially with n

Generalizes Faivre's result from beta=10 to all beta>1

Analyzes which expansion yields better approximations

Abstract

Let $\beta > 1$ be a real number and $x \in [0,1)$ be an irrational number. We denote by $k_n(x)$ the exact number of partial quotients in the continued fraction expansion of $x$ given by the first $n$ digits in the $\beta$-expansion of $x$ ($n \in \mathbb{N}$). It is known that $k_n(x)/n$ converges to $(6\log2\log\beta)/\pi^2$ almost everywhere in the sense of Lebesgue measure. In this paper, we improve this result by proving that the Lebesgue measure of the set of $x \in [0,1)$ for which $k_n(x)/n$ deviates away from $(6\log2\log\beta)/\pi^2$ decays to 0 exponentially as $n$ tends to $\infty$, which generalizes the result of Faivre \cite{lesFai97} from $\beta = 10$ to any $\beta >1$. Moreover, we also discuss which of the $\beta$-expansion and continued fraction expansion yields the better approximations of real numbers.