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

Abstract

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 the continued fraction expansion of an irrational number , for a non-decreasing function , we completely determine the size of the set F2(φ)=x∈[0,1):∃1⩽k≠l⩽n, ak(x)ak+1(x)⩾φ(n),al(x)al+1(x)⩾φ(n) for infinitely many n∈N in terms of Lebesgue measure and Hausdorff dimension.