On solution -free sets for simultaneous additive equations.

Abstract

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 <italic>s</italic> variables, where <italic>s</italic> &ge; 9. We show that if a subset A of the natural numbers restricted to the interval [1, <italic>N</italic>] 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 &cap; [1, <italic>N</italic>] grows no faster than a constant multiple of <italic>N</italic>/(log log <italic>N</italic>)<italic><super>-c </super></italic> as <italic>N</italic> &rarr; infinity, where <italic>c</italic> > 0 is an absolute constant. In particular, we show that the density of A in [1, <italic>N</italic>] tends to zero as <italic>N</italic> tends to infinity. We then generalise this approach to a system of <italic>k</italic> translation and dilation invariant diagonal polynomials of degrees 1,...,<italic>k</italic> with integer coefficients in <italic>s</italic> > <italic>s</italic>₀(<italic>k</italic>) variables, where <italic>s</italic>₀(<italic> k</italic>) is of order (2 + <italic>o</italic>(1))<italic>k</italic><super> 2</super>log <italic>k</italic>. We show that if the system of polynomials possesses non-singular real and <italic>p</italic>-adic solutions for all primes <italic>p</italic>, then a subset A of the natural numbers which satisfies a notion of pseudorandomness referred to by Gowers as uniformity of degree <italic>k</italic> 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 &cap; [1, <italic>N</italic>] grows no faster than a constant multiple of <italic>N</italic>/(log log <italic>N</italic>)<italic><super>-c </super></italic> as <italic>N</italic> &rarr; infinity, where <italic>c</italic> > 0 is a constant depending on <italic>k</italic>. 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.