Diophantine approximation by rational numbers of certain parity types

Abstract

For a given irrational number, we consider the properties of best rational approximations of given parities. There are three different kinds of rational numbers according to the parity of the numerator and denominator, say odd/odd, even/odd and odd/even rational numbers. We study algorithms to find best approximations by rational numbers of given parities and compare these algorithms with continued fraction expansions.

Figures (6)

• Figure 1: The parallelograms $P_{\mathscr B}(x,\mathbf v_{p/q})$ (left) and $P_{\mathscr S}(x,\mathbf v_{p/q})$ (right)
• Figure 2: Vectors of best approximations
• Figure 3: Intermediate convergents and best signed approximations of $x$.
• Figure 4: The Farey tessellation $\mathcal{F}$ and the cutting sequence along the geodesic.
• Figure 5: The tessellation given by the $\Delta$-images of $\mathbf T$. Each triangle is the image of the transformation written on it. The red lines, blue and green lines indicate the axes of the conjugates of the reflections of $H_1$, $H_0$ and $H_\infty$.

Theorems & Definitions (25)

• Definition 1.1
• Definition 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1
• proof
• Proposition 2.2
• proof : Proof of Theorem \ref{['thm1']}
• Lemma 3.1
• proof