Diophantine Approximation in Small Degree

Here, the exponent of q in the upper bound is optimal because, when ξ has bounded partial quotients, there is also a constant c > 0 such that |ξ − p/q| ≥ cq for all rational numbers p/q (see Chapter I of [14]). Define the height H(P ) of a polynomial P ∈ R[T ] as the largest absolute value of its coefficients, and the height H(α) of an algebraic number α as the height of its irreducible polynomial over Z. Then the above estimate may be generalized in the following two ways related respectively with Mahler’s and Koksma’s classifications of numbers. Consider a real number ξ which, for a fixed integer n ≥ 1, is not algebraic over Q of degree ≤ n. On one hand, an application of Dirichlet’s box principle shows that there exist infinitely many non-zero polynomials P ∈ Z[T ] such that