In this thesis we study distribution properties of values of certain polynomial maps at integral points. The first set of values that we investigate is the values of a linear map at integral points on a quadratic surface. we establish conditions sufficient to ensure that this set is dense and' then under stronger conditions we show that this set is equidistributed. The methods employed in this situation depend on the uniform distribution of unipotent flows on homogeneous spaces. In particular we use Ratner's theorems. The second set of values that we explore is the values of a quadratic form at integral points. We show that under certain dimension and rationality conditions the n-point correlations of this set behave in a way which is consistent with the behaviour of the correlations of uniformly distributed set of random numbers. The methods used in this setting are based on Fourier analysis and certain asymptotic estimates for theta series.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:658846 |
| Date | January 2014 |
| Creators | Sargent, Oliver |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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