Metrical properties for the product of consecutive partial quotients in Hurwitz continued fractions

Let [0;a(1)(z),a(2)(z),& mldr;] be the Hurwitz continued fraction expansion of a complex irrational number z is an element of F:={x+iy:x,y is an element of[-1/2,1/2)}, where a(n)(z) are Gaussian integers and |a(n)(z)|>= 2. This paper aims to study the growth rate of the product of consecutive partial quotients. Specifically, for any integer m >= 1, we provide a criterion for determining the Lebesgue measure of the set {z is an element of F:|a(n+1)(z)& ctdot;a(n+m)(z)|>psi(n) for infinitely many n}, where psi : N -> R+ is a positive function. Let h be a positive and continuous function on the compact set F & horbar;. We further obtain the Hausdorff dimension of the set {z is an element of F:|a(n+1)(z)& ctdot;a(n+m)(z)|>e(h(z)+& ctdot;+h(Tn-1z)) for infinitely many n} and show that this set has large intersection property, where T:F -> F is the Hurwitz map induced by Hurwitz continued fractions. This extends the main result of Bugeaud et al (2023 arXiv:2306.08254).