Glasnik Matematicki, Vol. 44, No.2 (2009), 323-331.

ON THE APPROXIMATION TO ALGEBRAIC NUMBERS BY ALGEBRAIC NUMBERS

Yann Bugeaud

Abstract.   Let n be a positive integer. Let ξ be an algebraic real number of degree greater than n. It follows from a deep result of W. M. Schmidt that, for every positive real number ε, there are infinitely many algebraic numbers α of degree at most n such that |ξ - α| < H(α)^{-n - 1 + ε}, where H(α) denotes the naive height of α. We sharpen this result by replacing ε by a function H ε(H) that tends to zero when H tends to infinity. We make a similar improvement for the approximation to algebraic numbers by algebraic integers, as well as for an inhomogeneous approximation problem.

2000 Mathematics Subject Classification.   11J68.

Key words and phrases.   Approximation by algebraic numbers, Schmidt Subspace Theorem.

DOI: 10.3336/gm.44.2.05

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