Pointwise order of generalized Hofstadter functions $G, H$ and beyond

Abstract

Hofstadter's $G$ function is recursively defined via $G(0)=0$ and then $G(n)=n-G(G(n-1))$. Following Hofstadter, a family $(F_k)$ of similar functions is obtained by varying the number $k$ of nested recursive calls in this equation. We establish here that this family is ordered pointwise: for all $k$ and $n$, $F_k(n) \le F_{k+1}(n)$. For achieving this, a detour is made via infinite morphic words generalizing the Fibonacci word. Various properties of these words are proved, concerning the lengths of substituted prefixes of these words and the counts of some specific letters in these prefixes. We also relate the limits of $\frac{1}{n}F_k(n)$ to the frequencies of letters in the considered words.

Figures (4)

• Figure 1.1: Plotting $F_1,F_2,\ldots,F_5$
• Figure 1.2: Infinite words $x_2,\ldots,x_5$ (with highest letter in red)
• Figure 5.1: The first $\alpha_k$ and $\beta_k$, positive roots of $X^k{+}X{-}1$ and $X^k{-}X^{k-1}{-}1$.
• Figure 7.1: The known $(F_k^j,\le)$ lattice, displayed here for $1\le k,j\le 5$.

Theorems & Definitions (50)

• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof
• Proposition 2.4
• proof
• Theorem 3.1
• proof