On the structure of the dimension spectrum for continued fraction expansions

We analyse the Hausdorff dimension spectrum of continued fraction expansions with coefficients restricted to infinite subsets of N. We prove that the set of powers P_q = {q^n : n in N} has full dimension spectrum for each integer q at least 2, answering a question by Chousionis, Leykekhman and Urbanski. On the other hand, we show that the dimension spectrum for P_q^* = {q^n : n in N} union {1} has many gaps and regions where it is nowhere dense. We also investigate the case where A is generated by a monomial, M_q = {n^q : n in N}. For M_q we prove that the dimension spectrum is full for q in {1, 2, 3, 4, 5}, and it has a gap for each q at least 6. Furthermore, we show that for q in {6, 7, 8} the dimension spectrum of M_q is the disjoint union of two nontrivial closed intervals, and for q in {9, 10} it is the disjoint union of three nontrivial closed intervals. For q at least 11, we show that the dimension spectrum of M_q consists of finitely many disjoint nontrivial closed intervals. The results concerning M_q extend existing results for q = 1 and q = 2. In our analysis we employ Perron-Frobenius (transfer) operators and numerical tools developed by Falk and Nussbaum that give rigorous estimates for the Hausdorff dimension of continued fraction expansions.