Consider algebraic integers α0, α1, . . . αd of degree d, not roots of unity, that are Galois conjugates. We wish to understand when one has |α αc1 ...αck|≥1wherek<d. Wecallthesetofalltuples(c ,...,c )∈ 01k 1k Rk≥0 for which this inequality holds Ek,d. We prove various properties of Ek,d and observing that it describes a polytope, we give explicit description of its members and describe its shape. We further explore the situation better bound in the approximation of algebraic numbers by rational numbers whose denominators form a non-degenerate, non-polynomial, linear recurrence sequence. Exploration of products of the form α αc1 . . . αck also 01k promises applications on linear forms in logarithms. This is joint work with Samprit Ghosh, Greg Knapp and Khoa Nguyen.