2021/07/10 by Nihar Gargava, Gargava, Nihar Prakash · 1 citation
Mathematics · Computer Science · #Mathematical Dynamics and Fractals #semigroups and automata theory #Limits and Structures in Graph Theory
paper · pdf · doi:10.48550/arxiv.2107.04844
In this article, we will show the existence of lattice packings in a sparse family of dimensions. This construction will be a generalisation of Venkatesh's lattice packing result. In our construction, we replace the appearance of the cyclotomic number field with a division algebra over the rational field. For this, we develop an analogue of Siegel's mean value theorem over lattices that have a prescribed set of symmetries given by a finite non-commutative group inside the multiplicative subgroup of a division algebra. This approach improves the best known lattice packing bounds in many dimensions.