Masters Thesis

Moduli space of dual pivots of a 3-D lattice

Abstract. Convergents of a real number can be analyzed by studying the pivots of a 2-D lattice and one can generalize this to analyze the convergents of a pair of real numbers by studying the pivots of a 3-D lattice. Therefore, studying the pivots of lattices gives us insights into problems in Diophantine approximation such as the Littlewood conjecture and Margulis conjecture. In this thesis, we characterize the type of 2-D sublattices that gives rise to pivots of the dual lattice of a 3-D lattice in R3, which we call dual pivots. We then apply this result to construct a moduli space of dual pivots that allows us to explore the relationships between pivots of a 3-D lattice and pivots of the corresponding dual lattice.

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