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 \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$$$ \end{document}, i.e., each number n\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$$$ \end{document} has a unique representation of the form n = c0 + c1 (−q) + ... + ch (−q)h, with ci\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\varepsilon$$ \end{document} {0,..., q − 1}(0 ≤ i ≤ h). The aim of this paper is to investigate the sum of digits function ν−q (n) of these number systems. In particular, we derive an asymptotic expansion for\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document} $$\sum\limits_{n < N} {|v_{ - q} (n)} - v_{ - q} ( - n)|$$ \end{document}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.