In a recent paper, Bugeaud and Nguyen proved a stronger version of a Theorem of Lenstra and Shallit related to the convergents of certain algebraic integers. One of the key ingredients of their proof was certain exponential relations observed among the absolute values of the Galois conjugates of an algebraic integer. In this talk we'll investigate whether the exponents in these relations are optimal and present a proof of similar Diophantine inequalities in a more general setting. This is a joint work with Greg Knapp, Khoa Nguyen and Seda Albayrak.