HAUSDORFF MEASURE AND LINEAR-FORMS

Given a dimension function f we prove that the Hausdorff measure H-f o f the set W(m, n; psi) of 'well approximable' linear forms is determin ed by the convergence or divergence of the sum (r=1)Sigma(infinity) f( psi(r))psi(r)(-(m-1)nrm+n-1). This is a Hausdorff measure analogue of the classical Khintchine-Groshev Theorem where the mn-dimensional Lebe sgue measure of W(m, n; psi) is determined by the convergence or diver gence of an mn-volume sum. Our results show that there is no dimension function for which H-f(W(m, n; psi)) is positive and finite.