Beta Expansions for Regular Pisot Numbers
Department of Pure Mathematics
University of Waterloo
Waterloo, ON N2L 3G1
A beta expansion is the analogue of the base 10 representation of a
real number, where the base may be a non-integer. Although the greedy
beta expansion of 1 using a non-integer base is, in general, infinitely
long and non-repeating, it is known that if the base is a Pisot number,
then this expansion will always be finite or periodic. Some work has
been done to learn more about these expansions, but in general these
expansions were not explicitly known. In this paper, we present a
complete list of the greedy beta expansions of 1 where the base is any
regular Pisot number less than 2, revealing a variety of remarkable
patterns. We also answer a conjecture of Boyd regarding cyclotomic
co-factors for greedy expansions.
Full version: pdf,
Received March 7 2011;
revised version received May 18 2011.
Published in Journal of Integer Sequences, June 2 2011.
Journal of Integer Sequences home page