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

TL;DR

This paper proves that a family of generalized Hofstadter functions is ordered pointwise, using properties of infinite morphic words related to Fibonacci words, and explores their limits and letter frequencies.

Contribution

It establishes the pointwise ordering of the family of functions $F_k$, extending Hofstadter's original recursive function, through analysis of morphic words and their properties.

Findings

The functions $F_k$ are ordered pointwise: $F_k(n) \\le F_{k+1}(n)$ for all $k$ and $n$.

Properties of infinite morphic words generalizing Fibonacci words are characterized.

Limits of $rac{1}{n}F_k(n)$ relate to letter frequencies in the associated morphic words.

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.