Mahler measures generate the largest possible groups

We prove that free additive and multiplicative groups generated by the set of all Mahler measures consist of all real algebraic integers and of all positive algebraic numbers, respectively. More precisely, we show that every positive algebraic number can be written as a quotient of two Mahler measures. It is also shown that the set of all Mahler measures is not an additive semigroup.

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Published 1 January 2004