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K. F. Roth

doi:10.1112/S0025579300000826

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On page 2 of my paper “Rational approximations to algebraic numbers” [Mathematika 2 (1955), 1–20], I referred to possible generalizations of my theorem, and mentioned that I had proved the following:

Let α be any algebraic number, not rational. If the inequality

is satisfied by infinitely many algebraic numbers β of degree g, then κ ≤ 2g.

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