Certain Diophantine properties of the Mahler measure
Let \(M(a)\) be the Mahler's measure of an algebraic number, and let \(\Omega=\{M(a):\;a\text{\;algebraic}\}\). Assume that the polynomial \(P\in{\mathbb Q}[X_1,\dots,X_n]\) has nonnegative coefficients, and depends on all its variables. It is shown that if \(a_1,\dots,a_n\in\Omega\) and \(P(a_1,\dots,a_n)\) is a positive rational integer, then all \(a_i\)'s are positive rational integers. The author obtains also a lower bound for the distance of \(a\in\Omega\setminus{\mathbb Z}\) from the nearest rational integer, and proves that if \(c_1,\dots,c_n\) are algebraic units, then for all \(k_1,k_2,\dots,k_n\in{\mathbb Z}\) the product \(\prod_{i=1}^nM(c_i)^{k_i}\) is either irrational, or equal to \(1\).
scientific article; zbMATH DE number 1928343