On Numbers which are Mahler Measures

Abstract

We investigate which algebraic numbers can be Mahler measures. Adler and Marcus showed that these must be Perron numbers. We prove that certain integer multiples of every Perron number are Mahler measures. The results of Boyd give some necessary conditions on Perron number to be a measure. These do not include reciprocal algebraic integers, so it would be of interest to find one which is not a Mahler measure. We prove a result in this direction. Finally, we show that for every non-negative integer k there is a cubic algebraic integer having norm 2 such that precisely the kth iteration of its Mahler measure is an integer.