Journal of Integer Sequences, Vol. 14 (2011), Article 11.6.4

Beta Expansions for Regular Pisot Numbers

Maysum Panju
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.

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Received March 7 2011; revised version received May 18 2011. Published in Journal of Integer Sequences, June 2 2011.

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