Generalized Hofstadter functions $G, H$ and beyond: numeration systems and discrepancy
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866910908786147328 |
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| author | Letouzey, Pierre |
| author_facet | Letouzey, Pierre |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_12615 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Generalized Hofstadter functions $G, H$ and beyond: numeration systems and discrepancy Letouzey, Pierre Discrete Mathematics Formal Languages and Automata Theory Logic in Computer Science Combinatorics Number Theory 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. |
| title | Generalized Hofstadter functions $G, H$ and beyond: numeration systems and discrepancy |
| topic | Discrete Mathematics Formal Languages and Automata Theory Logic in Computer Science Combinatorics Number Theory |
| url | https://arxiv.org/abs/2502.12615 |