Title:
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Shifts, averages and restriction of forms in several variables
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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.
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