Odd-odd continued fraction algorithm

Abstract

By using a jump transformation associated to the Romik map, we define a new continued fraction algorithm called odd-odd continued fraction, whose principal convergents are rational numbers of odd denominators and odd numerators. Among others, it is proved that all the best approximating rationals of odd denominators and odd numerators of an irrational number are given by the principal convergents of the odd-odd continued fraction algorithm and vice versa.

Figures (4)

• Figure 1: The graph of $R$ (left) and the graph of $T_\mathrm{OOCF}$ (right)
• Figure 2: Ford circles: white circles are based at $\infty$-rationals and gray circles are based at $1$-rationals
• Figure 3: Two possible relative locations of $x$, $p_n/q_n$, $a/b$ and $p"_n/q"_n$ in the proof of Theorem \ref{['Thm:main1']}. The dashed circles are the horocycles based at $x$ tangent to $C_{p_n/q_n}$ and $C_{a/b}$.
• Figure 4: A possible relative position of $x$, $a/b$ and the convergents. The dashed circles $C$ and $C'$ are horocycles based at $x$ tangent to $C_{a/b}$ and $C_{p_{n-1}/q_{n-1}}$.

Theorems & Definitions (28)

• Definition 1.1
• Theorem 1.2
• Theorem 1.3
• Proposition 2.1
• Proposition 2.2
• Lemma 3.1
• Proposition 3.2
• Lemma 3.3
• Theorem 3.4
• proof