On Noncanonical Number Systems

Christiaan van de Woestijne

Technische Universität Graz, Austria

We propose a way to extend the definition of Canonical Number Systems (CNS) to general finite free $\mathbb{Z}$-modules and to general digit sets that need not, for example, contain the zero digit. In this setting, we prove quite generally that if all eigenvalues of the base of the system are greater than $2$ in absolute value, there always exists a digit set that allows finite expansions of all elements in the module. We also give a brief exploration of some number systems with all digits nonzero, as well as a ``product operation'' on number systems and their digit sets using the Chinese Remainder Theorem.

Supported by the Austrian Science Fundation (FWF), project nb S9606.