On The Sum of Digits Function for Number Systems with Negative Bases

Let q 2 be an integer. Then –q gives rise to a number system in , i.e., each number n has a unique representation of the form n = c ₀ + c ₁ (–q) + ... + c _h (–q) ^h , with c _i {0,..., q – 1}(0 i h). The aim of this paper is to investigate the sum of digits function _–q (n) of these number systems. In particular, we derive an asymptotic expansion forand obtain a Gaussian asymptotic distribution result for _–q (n) – _–q (–n). Furthermore, we prove non-differentiability of certain continuous functions occurring in this context. We use automata and analytic methods to derive our results.