Fast Arithmetical Algorithms in Möbius Number Systems

We analyze the time complexity of exact real arithmetical algorithms in Möbius number systems. Using the methods of Ergodic theory, we associate to any Möbius number system its transaction quotient T ≥ 1 and show that the norm of the state matrix after n transactions is of the order Tⁿ. We argue that the Bimodular Möbius number system introduced in Kůrka [10] has transaction quotient less than 1.2, so that it computes the arithmetical operations faster than any standard positional system.