| In this thesis we investigate two non-linear problems from arithmetic combinatorics by means of a variant of the Hardy-Littlewood circle method. We first consider a translation and dilation invariant system consisting of a diagonal quadratic equation and a linear equation with integer coefficients in s variables, where s ≥ 9. We show that if a subset A of the natural numbers restricted to the interval [1, N] satisfies a notion of pseudorandomness which Gowers terms quadratic uniformity, then it furnishes roughly the expected number of simultaneous solutions to the given equations. If A furnishes no non-trivial solutions to the given system, then we show that the number of elements in A ∩ [1, N] grows no faster than a constant multiple of N/(log log N)-c as N → infinity, where c > 0 is an absolute constant. In particular, we show that the density of A in [1, N] tends to zero as N tends to infinity.; We then generalise this approach to a system of k translation and dilation invariant diagonal polynomials of degrees 1,...,k with integer coefficients in s > s 0(k) variables, where s0( k) is of order (2 + o(1))k 2log k. We show that if the system of polynomials possesses non-singular real and p-adic solutions for all primes p, then a subset A of the natural numbers which satisfies a notion of pseudorandomness referred to by Gowers as uniformity of degree k furnishes roughly the expected number of simultaneous solutions to the given system. If A furnishes no non-trivial solutions to the given system, then we show that the size of A ∩ [1, N] grows no faster than a constant multiple of N/(log log N)-c as N → infinity, where c > 0 is a constant depending on k.; These results, which generalise earlier work of Roth on systems of translation and dilation invariant systems of linear equations, mark the first study of solution-free sets for systems which do not consist purely of linear equations, and the analysis of which is dominated by the non-linear components. |
