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Spiraling of lattice approximates

Jim Tseng
(University of Bristol)
Torsdag, 6 december, 2012, at 16:15-17:15, in Aud. D3 (1531-215)
A classical theorem of Dirichlet from the theory of Diophantine approximation, a subfield of number theory, says that, for every real d-vector x, there exists infinitely many pairs of (p,q) satisfying ||qx - p|| < |q|^(-1/d) where p is a d-vector with integer components, q is a nonzero integer, and ||*|| is the distance to the nearest integer lattice point.

One can regard these pairs (p,q) as lattice points in a thinning region of a unimodular lattice associated to x. Take the orthogonal projection of these lattice points in this region. For each resulting d-vector, further project radially onto the unit d-1-sphere (which lies in the orthogonal projection) and consider the distribution of such vectors. One can now ask for this distribution for an analogous thinning region for any unimodular lattice. We show that, on average, the distribution is uniform - i.e. on average, the directions of such orthogonally projected lattice points are uniformly distributed.

Conversely, we construct specific examples for which the distribution is non-uniform.

This is joint work with J. Athreya and A. Ghosh.
Kontaktperson: Simon Kristensen