Configurations of Rank-40r Extremal Even Unimodular Lattices (r=1,2,3)

We show that if L is an extremal even unimodular lattice of rank 40r with r=1,2,3 then L is generated by its vectors of norms 4r and 4r+2. Our result is an extension of Ozeki's result for the case r=1.Comment: 5 pages. To appear, Journal de Theorie des Nombres de Bordeau

Coincidences of a shifted hexagonal lattice and the hexagonal packing

A geometric study of twin and grain boundaries in crystals and quasicrystals is achieved via coincidence site lattices (CSLs) and coincidence site modules (CSMs), respectively. Recently, coincidences of shifted lattices and multilattices (i.e. finite unions of shifted copies of a lattice) have been investigated. Here, we solve the coincidence problem for a shifted hexagonal lattice. This result allows us to analyze the coincidence isometries of the hexagonal packing by viewing the hexagonal packing as a multilattice.Comment: 8 pages, 2 figures, submitted to ICQ12 Conference Proceeding

LpL^{p} and Weak-LpL^{p} estimates for the number of integer points in translated domains

Revisiting and extending a recent result of M.Huxley, we estimate the $L^{p}\left( \mathbb{T}^{d}\right)$ and Weak-$L^{p}\left( \mathbb{T}^{d}\right)$ norms of the discrepancy between the volume and the number of integer points in translated domains