Multiplicative dependence of shifted algebraic numbers
We show that the set obtained by adding all sufficiently large integers to a fixed quadratic algebraic number is multiplicatively dependent. So also is the set obtained by adding rational numbers to a fixed cubic algebraic number. Similar questions for algebraic numbers of higher degrees are also raised. These are related to the Prouhet–Tarry–Escott type problems and can be applied to the zero-distribution and universality of some zeta-functions.