Given $i_1<\ldots <i_k$, t_1,\ldots, t_k>0$, and $B>0$, we consider the set $\{x; \prod_{\ell=1}^k a_{n+i_\ell}^{t_\ell}(x)> B^n {\rm i.o.}\}$ and calculate its Hausdorff dimension. It is a generalization of results of Huang, Wu, Xu, and Bakhtawar, Hussein, Kleinbock, Wang.

To make the result more general, we consider not continuous fractions but a more general setting of $d$-decaying Gauss-like systems. We do not assume any distortion bounds, so the first step is to prove the basic thermodynamical formalism (in particuler, existence of pressure for the geometric potentials) for such systems.

It is a joint work with Ayreena Bakhtawar.