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Title: Shifts, averages and restriction of forms in several variables
Author: Chow, Sam
ISNI:       0000 0004 6059 7136
Awarding Body: University of Bristol
Current Institution: University of Bristol
Date of Award: 2016
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Our main focus shall be the use of Fourier analytic methods to count solutions to diophantine equations and inequalities. We begin by using the Hardy-Littlewood circle method to produce mean and variance statistics for the number of solutions to diophantine equations in a thin family. The bulk of this thesis concerns the study of diophantine inequalities, III particular using the Davenport- Heilbronn method. In many cases inequalities may be treated analogously to equations, but sometimes new difficulties arise. Initially, we consider the special case in which a cubic diophantine inequality splits into several parts, providing lower bounds for the number of variables required to ensure that the inequality has a nontrivial solution. Research on diophantine inequalities has previously focussed on homogeneous polynomials with real coefficients. We investigate a new type of inequality problem, involving rational polynomials evaluated at irrationally shifted copies of the integers. The diagonal case gives rise to a new inequalities analogue to Waring's problem, in which sums of shifted powers are considered. Moving onto more general systems of polynomials, we present the first inequalities analogue to Birch's theorem. When it comes to diophantine equations, a popular objective is to demonstrate that an equation has the expected number of solutions. For cubic equations, we supplement the existing literature by showing that these solutions are evenly distributed, in a precise sense. The analytic methods presented here generalise to arbitrary degree. Finally, we consider Waring-Goldbach equations in which the variables are restricted to lie in a prescribed set. Specifically, we show that any subset of the dth powers of primes with positive relative density contains nontrivial solutions to a translation-invariant linear equation in d2 + 1 or more variables, with explicit quantitative bounds.
Supervisor: Not available Sponsor: Not available
Qualification Name: Thesis (Ph.D.) Qualification Level: Doctoral
EThOS ID:  DOI: Not available