Lower rational approximations and Farey staircases

Abstract

For a real number $x$, call $\frac1n \lfloor nx \rfloor$ the $n$-th lower rational approximation of $x$. We study the functions defined by taking the cumulative average of the first $n$ lower rational approximations of $x$, which we call the Farey staircase functions. This sequence of functions is monotonically increasing. We determine limit behavior of these functions and show that they exhibit fractal structure under appropriate normalization.

Figures (7)

• Figure 1: Lower rational approximations $\frac{1}{n} \left\lfloor{nx}\right\rfloor$, for $n = 3,4,5$.
• Figure 2: Farey staircases $A_n(x)$ for $n = 3, 4, 5$.
• Figure 3: Incremented staircases $D_n(x)$ for $n = 3, 4, 5$.
• Figure 4: Farey staircase $A_{30}(x)$ on the domain $[0,1]$.
• Figure 5: Incremented staircase $D_{30}(x)$, normalized to height one.

Theorems & Definitions (23)

• Theorem 1
• Remark 2
• Theorem 3
• Theorem 4
• Theorem 5: see richman
• Corollary 6: to Theorem \ref{['thm:limit-main']}
• Proposition 7
• proof
• Lemma 8
• proof