The parity of the Zeckendorf sum-of-digits function

Abstract:

Let s(n) denote the sum of digits of the Zeckendorf representation of n and . The aim of this paper is to discuss the behaviour of $S_{q,i}(N)$. First it is shown that that the values of admit a Gaussian limit law with bounded mean and variance of order log N. Conversely, for q≤1 (mostly) has a periodic fractal structure. We also prove that which is an analogue to a well-known result by Newman [14] for binary digit expansions.