A note on the relative growth of products of multiple partial quotients in the plane

Let r = [a(1)(r), a(2)(r), ...] be the continued fraction expansion of a real number r epsilon R. The growth properties of the products of consecutive partial quotients are tied up with the set admitting improvements to Dirichlet's theorem. Let (t(1), ..., t(m)) epsilon R-+(m), and let Psi : N -> (1, infinity) be a function such that Psi(n) -> infinity as n -> infinity. We calculate the Hausdorff dimension of the set of all (x, y) is an element of [0,1)(2) such that max{Pi(m)(i=1) a(n+i)(ti)(x), Pi(m)(i=1) a(n+i)(ti)(y)} >= Psi(n) is satisfied for all n >= 1.