Pure Maths Colloquium: Luke Jeffreys

Initially studied by Markov around 1880, the Lagrange and Markov spectra are complicated subsets of the real line that play a crucial role in the study of Diophantine approximation and the study of binary quadratic forms. Perron’s description of the spectra in terms of continued fractions allowed the machinery of Gauss-Cantor sets (Cantor sets related to the dynamics of the Gauss map on continued fractions) to come to bear on many problems. In the 1960s, Freiman demonstrated that the Lagrange spectrum is a strict subset of the Markov spectrum. It still remains a difficult task to find points in the complement of the Lagrange spectrum within the Markov spectrum and modern research is focussed on further developing our understanding of this complement.

In this talk, I will introduce these spectra, discussing the historical results above, and speak about recent work with Carlos Matheus and Carlos Gustavo Moreira finding new points in the complement and obtaining better heuristic lower bounds for its Hausdorff dimension.