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Title: Quantitative results in arithmetic combinatorics
Author: Bloom, Thomas F.
ISNI:       0000 0004 5370 5370
Awarding Body: University of Bristol
Current Institution: University of Bristol
Date of Award: 2014
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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.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available