Combinatorial constructions for the Zeckendorf sum of digits of polynomial values

Abstract

Let \(p(X)\in\mathbb{Z}[X]\) with \(p(\mathbb{N})\subset\mathbb{N}\) be of degree h≥2 and denote by sF(n) the sum of digits in the Zeckendorf representation of n. We study by combinatorial means three analogues of problems of Gelfond (Acta Arith. 13:259–265, 1967/1968), Stolarsky (Proc. Am. Math. Soc. 71:1–5, 1978) and Lindstrom (J. Number Theory 65:321–324, 1997) concerning the distribution of sF on polynomial sequences. First, we show that for m≥2 we have #{n<N:sF(p(n))≡amodm}≫p,mN4/(6h+1) (Gelfond). Secondly, we find the extremal minimal and maximal orders of magnitude of the ratio sF(p(n))/sF(n) (Stolarsky). Third, we prove that lim supn→∞sF(p(n))/logφ(p(n))=1/2, where φ denotes the golden ratio (Lindstrom).