On sums of partial quotients in Hurwitz continued fraction expansions (Q6167188)

Assume that a complex number \(z = x+ iy\) has its Hurwitz continued fraction expansion \([0; a_1(z), a_2(z),\dots.]\). It is proved that the sets of \(z\) for which accumulation points of the sequence of partial arithmetic means of \(a\)'s attains two prescribed complex values is of full Hausdorff dimension. A similar result is provided for Hausdorff dimension of a set of \(z\), for which the cluster set of a sequence of partial arithmetic means of absolute values of \(a\)' s lies in a prescribed interval.