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

Abstract

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 = c0 + c1 (−q) + ... + c h (−q) h , with c i \(\varepsilon\) {0,..., q − 1}(0 ≤ i ≤ h). The aim of this paper is to investigate the sum of digits function ν−q (n) of these number systems. In particular, we derive an asymptotic expansion for $$\sum\limits_{n < N} {|v_{ - q} (n)} - v_{ - q} ( - n)|$$ and obtain a Gaussian asymptotic distribution result for ν−q(n) − ν−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.