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In this note we will describe a simple and practical approach to get rigorousbounds on the Hausdorff dimension of limits sets for some one dimensionalMarkov iterated function schemes. The general problem has attractedconsiderable attention, but we are particularly concerned with the role of thevalue of the Hausdorff dimension in solving conjectures and problems in otherareas red of mathematics. As our first application we confirm, and oftenstrengthen, conjectures on the difference of the Lagrange and Markov spectra inDiophantine analysis, which appear in the work of Matheus and MoreiraarXiv:1803.01230. As a second application we (re-)validate and improveestimates connected with the Zaremba conjecture in number theory, used in thework of Bourgain-Kontorovich arXiv:1107.3776v2, Huang arXiv:1310.3772v4 and KanarXiv:1604.04884. As a third more geometric application, we rigorously boundthe bottom of the spectrum of the Laplacian for infinite area surfaces, asillustrated by an example studied by McMullen. In all approaches to estimatingthe dimension of limit sets there are questions about the efficiency of thealgorithm, the computational effort required and the rigour of the bounds. Theapproach we use has the virtues of being simple and efficient and we present itin section 3 in a way that is straightforward to implement.

Hausdorff dimension estimates applied to Lagrange and Markov spectra, Zaremba theory, and limit sets of Fuchsian groups