On General 2-dimensional Lattice Spectra: Closedness, Hall's Ray, and Examples

TL;DR

This paper develops new techniques to analyze 2D lattice spectra, unifies several famous spectra through a log-systole function, and proves properties like closedness and Hall's interval existence across different applications.

Contribution

It introduces a unifying log-systole function for multiple spectra, computes the Mordell-Gruber spectrum in 2D, and generalizes Perron's formulas and Cantor set sums.

Findings

Computed the Mordell-Gruber spectrum in 2D

Proved closedness of certain spectra

Established existence of Hall's interval in various cases

Abstract

The Lagrange and Markov spectra have been studied since late 19th century, concerning badly approximable real numbers. The Mordell-Gruber spectrum has been studied since 1936, concerning the supremum of the area of a rectangle centered at the origin that contains no other points of a unimodular lattice. We develop techniques that incorporate unimodular lattices and integer sequences, providing the log-systole function which unifies four famous spectra. We compute the Mordell-Gruber spectrum in the two-dimensional case and generalize Perron's formulas behind some famous spectra. Furthermore, we generalize the sum of Cantor sets to prove that certain functions on cartesian product of two Cantor sets contain an interval. Combining the techniques, we prove closedness and existence of Hall's interval in several different applications.