Beta representations with minimal average digit

Abstract

We consider the problem of minimising the average digit in beta representations with unrestricted digits, equivalent to a one dimensional affine switched system obtained from a formal limit of a stabilisable two dimensional linear system. We give a countable set of ${\hat{I}}^2$ for which the result is given by the usual (greedy) beta expansion, an interval of values for which it is strictly less, a numerical upper bound, and a conditional lower bound for all ${\hat{I}}^2$.

15 pages, 1 figure. Significantly repackaged, including a new title, definitions and structure