Apparaît dans la collection : Combinatorics and Arithmetic for Physics : Special Days
In 1945, Siegel showed that the expected value of the lattice-sums of a function
over all the lattices of unit covolume in an n-dimensional real vector space is
equal to the integral of the function. In 2012, Venkatesh restricted the lattice-
sum function to a collection of lattices that had a cyclic group of symmetries and
proved a similar mean value theorem. Using this approach, new lower bounds
on the most optimal sphere packing density in n dimensions were established for
infinitely many n.
In the talk, we will outline some analogues of Siegel’s mean value theorem
over lattices. This approach has modestly improved some of the best known lattice
packing bounds in many dimensions. We will also show how such results can be
made effective and talk of some variations.
(Joint work with Vlad Serban.)