ON A THEOREM OF WIRSING IN DIOPHANTINE APPROXIMATION

Let n and d be integers with 1 <= d <= n-1. Let xi be a real number which is not algebraic of degree at most n. We establish that there exist an effectively computable constant c, depending only on xi and on n, an integer k with 1 <= k <= d, and infinitely many integer polynomials P(X) of degree m at most equal to n whose roots alpha(1), ..., alpha(m) can be numbered in such a way that |(xi - alpha(1)) ... (xi - alpha(k))| <= cH(P)(-) (d/d+1) (n - 1/d+1-1). This extends a well-known result of Wirsing who dealt with the case d = 1.