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

TL;DR

The paper addresses generalized Hofstadter functions $F_k$ and their digital-expansion interpretations, seeking how closely $F_k(n)$ tracks linear forms $\alpha_k n$ with $\alpha_k$ the positive root of $X^k+X-1$. It develops Fibonacci-like $A_{k,p}$-based numeration systems, establishes Zeckendorf-type decompositions, and relates $F_k$ to shifts on these decompositions via the morphic-words framework, all while analyzing the associated polynomials $P_k$ and $Q_k$ and their zeros. The main results show finite discrepancy $\Delta_k$ iff $k\le4$ (with explicit bounds $\Delta_3\approx0.854$ and $\Delta_4\approx1.58$) and infinite discrepancy for $k\ge5$, including a logarithmic growth at $k=5$ and a sublinear growth $n^a$ for $k\ge6$; applications resolve longstanding OEIS conjectures for $F_3$ and $F_4$ and reveal a Rauzy-like fractal structure in the discrepancy. The work combines combinatorial constructions, algebraic-number theory, and Coq-based formal verification to illuminate the interplay between digital representations and Hofstadter-type dynamics, with implications for numeration systems and symbolic dynamics.

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.

Figures (4)

• Figure 1.1: Plotting $F_1,F_2,\ldots,F_5$.
• Figure 3.1: Initial values of $A_{k,p}$.
• Figure 8.1: A Jacobi-Perron fractal: displaying $(\delta_3(n),\delta_3(F_3(n)))$ for $n<10^4$.
• Figure 9.1: Summary of results.

Theorems & Definitions (93)

• Definition 1.1
• Definition 1.2
• Proposition 2.1
• Proposition 2.2
• Proposition 2.3
• proof
• Corollary 2.4
• proof
• Definition 3.1
• Proposition 3.2