Sets of exact 'logarithmic' order in the theory of Diophantine approximation

For each real number alpha, let E(alpha) denote the set of real numbers wit h exact order alpha. A theorem of Guting states that for alpha greater than or equal to 2 the Hausdorff dimension of E(alpha) is equal to 2/alpha. In this note we introduce the notion of exact t-logarithmic order which refine s the usual definition of exact order. Our main result for the associated r efined sets generalizes Guting's result to linear forms and moreover determ ines the Hausdorff measure at the critical exponent. In fact, the sets are shown to satisfy delicate zero-infinity laws with respect to Lebesgue and H ausdorff measures. These laws are reminiscent of those satisfied by the cla ssical set of well approximable real numbers, for example as demonstrated b y Khintchine's theorem.