  In this note we will describe a simple and practical approach to get rigorous
bounds on the Hausdorff dimension of limits sets for some one dimensional
Markov iterated function schemes. The general problem has attracted
considerable attention, but we are particularly concerned with the role of the
value of the Hausdorff dimension in solving conjectures and problems in other
areas red of mathematics. As our first application we confirm, and often
strengthen, conjectures on the difference of the Lagrange and Markov spectra in
Diophantine analysis, which appear in the work of Matheus and Moreira
arXiv:1803.01230. As a second application we (re-)validate and improve
estimates connected with the Zaremba conjecture in number theory, used in the
work of Bourgain-Kontorovich arXiv:1107.3776v2, Huang arXiv:1310.3772v4 and Kan
arXiv:1604.04884. As a third more geometric application, we rigorously bound
the bottom of the spectrum of the Laplacian for infinite area surfaces, as
illustrated by an example studied by McMullen. In all approaches to estimating
the dimension of limit sets there are questions about the efficiency of the
algorithm, the computational effort required and the rigour of the bounds. The
approach we use has the virtues of being simple and efficient and we present it
in section 3 in a way that is straightforward to implement.
