Given $i_1<\ldots<i_k$, $B>1$, and $t_1,\ldots,t_k>0$, we consider the set of points whose continuous fraction expansions $(a_i(x))_i$ satisfy $\prod_{i=1}^k a_{n+a_i}^{t_i} \geq B^n$ for infinitely many $n$, and calculate its Hausdorff dimension. Several simpler versions of this problem were already considered in the literature (and even presented on our seminar), the motivation for studying this kind of sets goes back to a paper by Kleinbock and Wadleigh.

We actually consider a more general version of this question, for so-called $d$-decaying Gauss-like systems (introduced by Liao and myself as a generalization of the Gauss map and all its standard modifications). To do this we need to prove some basic results from thermodynamical formalism for this class of systems (the thermodynamical formalism for infinite IFS created by Mauldin and Urbański uses certain condition, which they call acceptability, and which is not satisfied in our case).

Meeting ID: 852 4277 3200 Passcode: 103121