Simultaneous approximation of a real number by all conjugates of an algebraic number
Guillaume Alain (2007)
Acta Arithmetica
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Guillaume Alain (2007)
Acta Arithmetica
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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.