Generalized Hostadter functions G, H and beyond: numeration systems and discrepancy
Résumé
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. The Fibonacci-like sequences are then expressed in terms of zeros of the polynomial X^k -X^(k-1) -1. Thanks to that, we estimate the discrepancy of each function F_k , i.e., the maximal distance between F_k and its linear equivalent. This discrepancy is finite exactly when k ≤ 4. This also solves some related questions, in particular 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 F_k can coincide exactly with such an integer part only when k ≤ 2, while F_k is almost additive exactly when k ≤ 4. Finally, a nice fractal shape a la Rauzy has been encountered when investigating the discrepancy of F_3 .
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