Dirichlet's diophantine approximation theorem

One form of Dirichlet's theorem on simultaneous diophantine approximation asserts that if α1, α2, …, αn are any real numbers and m ≥ 2 is an integer, then there exist integers q, p1, p2, …, pn such that 1 ≤ q < m and |qαi.-pi| ≤ m–1/n holds for 1 < i < n. The paper considers the problem of the extent to which this theorem can be improved by replacing m–1/n by a smaller number. A general solution to this problem is given. It is also shown that a recent result of Kurt Mahler [Bull. Austral. Math. Soc. 14 (1976), 463–465] amounts to a solution of the case n = 1 of the above problem. A related conjecture of Mahler is proved.