Metrical properties of the large products of partial quotients in continued fractions

The study of products of consecutive partial quotients in the continued fraction arises naturally out of the improvements to Dirichlet's theorem. We study the distribution of the two large products of partial quotients among the first n terms. More precisely, writing [a1(x),a2(x), horizontal ellipsis ] the continued fraction expansion of an irrational number x is an element of(0,1) , for a non-decreasing function phi:N -> R , we completely determine the size of the set F2(phi)=x is an element of[0,1):there exists 1 <= k not equal l <= n, ak(x)ak+1(x)>phi(n),al(x)al+1(x)>phi(n) for infinitely many n is an element of N in terms of Lebesgue measure and Hausdorff dimension.