Title:

Spherical equidistribution in the space of adelic lattices and its applications

The cutandproject construction is a way of producing "quasiperiodic"
point sets in Rd. We start by fixing a lattice in Rd x Rm for some integer
m >/1. We then choose a "nice" set W in Rm and project onto ]Rd those
lattice points whose projections onto Rm lie in W.
By placing a spherical scatterer around each of these points and considering
the ballistic motion of a point particle in this array of obstacles,
one obtains a Lorentz gas model with a quasiperiodic scatterer configuration.
In this setting, Jens Marklof and Andreas Strombergsson proved the
existence of a limiting distribution of free path lengths in the BoltzmannGrad
limit, complementing previous work concerning random and periodic
scatterer configurations.
The construction described above can be abstracted such that we may
replace Rm with some other locally compact abelian group. In this thesis,
we replace Rm with Ad/f where Af is the ring of finite adeles. This enables
us to' produce a number of arithmetically interesting point sets, such as
the primitive lattice points in Zd, and study the distribution of free path
lengths in the corresponding scatterer configurations.
The work of Marklof and Strombergsson 'crucially relies on W having
nonempty interior and boundary of measure O. In order to generate such
sparse sets as the primitive lattice points, both of those conditions must be '
removed. By proving a spherical equidistribution theorem on the space of
adelic lattices and a: general probabilistic result, we are nonetheless able to
deduce the existence of a limiting distribution for the free path lengths. By
the same token, we also derive results about the local statistics of directions
in translate's of these point sets. When d = 2, we prove the existence of a
universal limiting gap distribution for nonrational translates and that the
pair correlation for Diophantine translates is Poissonian.
