Hausdorff dimension for the weighted products of multiple digits in d-decaying Gauss like systems

Abstract

We compute the Hausdorff dimension of sets defined by the growth of weighted products of multiple digits at arbitrary positions in $d$-decaying Gauss-like iterated function systems. We provide the complete Hausdorff dimensional result for product of more than two digits, which was an open problem even for consecutive digits in the classical Gauss map and L\"{u}roth map. In our approach we do not need to assume the Bounded Distortion Property (BDP).