Generalized Hofstadter functions $G, H$ and beyond: numeration systems and discrepancy

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 study here some Fibonacci-like sequences that are deeply connected with these functions $F_k$. In particular, the Zeckendorf theorem can be adapted to provide digital expansions via sums of terms of these sequences. On these digital expansions, the functions $F_k$ are acting as right shifts of the digits. These Fibonacci-like sequences can be expressed in terms of zeros of the polynomial $X^k{-}X^{k-1}{-}1$. Considering now the discrepancy of each function $F_k$, i.e., the maximal distance between $F_k$ and its linear equivalent, we retrieve the fact that this discrepancy is finite exactly when $k \le 4$. Thanks to that, we solve two twenty-year-old OEIS conjectures stating how close the functions $F_3$ and $F_4$ are from the integer parts of their linear equivalents. Moreover we establish that $F_k$ can coincide exactly with such an integer part only when $k\le 2$, while $F_k$ is almost additive exactly when $k \le 4$. Finally, a nice fractal shape a la Rauzy has been encountered when investigating the discrepancy of $F_3$. Almost all this article has been formalized and verified in the Coq/Rocq proof assistant.