Combinatorial constructions for the Zeckendorf sum of digits of polynomial values
Constructions combinatoires pour la fonction sommes des chiffres de Zeckendorf des valeurs polynomiales
Résumé
Let p(X) ∈ Z[X] be of degree h ≥ 2 and denote by s_F (n) the sum of digits in the Zeckendorf representation of n. We study by combinatorial means three analogues of problems of Gelfond (1967/68), Stolarsky (1978) and Lindström (1997) concerning the distribution of s_F on polynomial sequences. First, we show that for m ≥ 2 we have #{n < N : s_F (p(n)) ≡ a mod m} p,m >> N^(4/(6h+1)) (Gelfond). Secondly, we find the extremal minimal and maximal orders of magnitude of the ratio s_F (p(n))/s_F (n) (Stolarsky). Third, we prove that lim sup n→∞ s_F(p(n))/ log_ϕ(p(n)) = 1/2, where ϕ denotes the golden ratio (Lindström).
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