In this thesis we study the generalisation of Roth's theorem on three term arithmetic progressions to arbitrary discrete abelian groups and translation invariant linear equations. We prove a new structural result concerning sets of large Fourier coefficients and use this to prove new quantitative bounds on the size of finite sets which contain only trivial solutions to a given translation invariant linear equation. In particular, we obtain a quantitative improvement for Roth's theorem on three term arithmetic progressions and its analogue over lFq[t], the ring of polynomials in t with coefficients in a finite field Fq. We prove arithmetic inverse results for lFq[t] which characterise finite sets A such that IA + t . AI / IAI is small. In particular, when IA + AI « IAI we prove a quantitatively optimal result, which is the lFq[t]-analogue of the Polynomial Freiman-Ruzsa conjecture in the integers. In joint work with Timothy G. F. Jones we prove new sum-product estimates for finite subsets of lFq[t], and more generally for any local fields, such as Qp. We give an application of these estimates to exponential sums.
| Identifer | oai:union.ndltd.org:bl.uk/oai:ethos.bl.uk:678687 |
| Date | January 2014 |
| Creators | Bloom, Thomas F. |
| Publisher | University of Bristol |
| Source Sets | Ethos UK |
| Detected Language | English |
| Type | Electronic Thesis or Dissertation |
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