The parity of the Zeckendorf sum-of-digits function (Q1974142)

The authors consider the quantity \(S(N)= \sum_{n<N} (-1)^{s(n)}\), where \(s(n)\) denotes the sum of digit functions with respect to the Zeckendorf expansion of \(n\) (i.e. the base sequence is given by the Fibonacci numbers). The authors prove a Gaussian limit law for \(S(N)\) with bounded mean and variance of order \(\log N\). Furthermore the authors obtain some results for similar digital sums, where the summation is extended over all \(n< N\) with \(n\equiv i\bmod q\) (\(q\) an integer \(>1\) and \(i\in \{0,\dots, q-1\}\)). In particular, they show the positivity of such sums for \(q=3\) and \(i=0\), which is an analogue of a well-known result by Newman for standard radix expansions.