The cut-and-project 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 non-empty 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 non-rational translates and that the pair correlation for Diophantine translates is Poissonian.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:702923 |
| Date | January 2016 |
| Creators | El-Baz, Daniel |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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