Exceptional sets for Diophantine inequalities

We apply Freeman's variant of the Davenport-Heilbronn method to investigate the exceptional set of real numbers not close to some value of a given real diagonal form at an integral argument. Under appropriate conditions, we show that the exceptional set in the interval [-N,N] has measure O(N^{1-c}), for a positive number c

Near-optimal mean value estimates for multidimensional Weyl sums

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the best possible. Several applications of our bounds are discussed

On Pairs of Diagonal Quintic Forms

We demonstrate that a pair of additive quintic equations in at least 34 variables has a nontrivial integral solution, subject only to an 11-adic solubility hypothesis. This is achieved by an application of the Hardy–Littlewood method, for which we require a sharp estimate for a 33.998th moment of quintic exponential sums. We are able to employ p -adic iteration in a form that allows the estimation of such a mean value over a complete unit square, thereby providing an approach that is technically simpler than those of previous workers and flexible enough to be applied to related problems.Peer Reviewedhttp://deepblue.lib.umich.edu/bitstream/2027.42/42604/1/10599_2004_Article_334960.pd

On generating functions in additive number theory, II: lower-order terms and applications to PDEs

We obtain asymptotics for sums of the formSigma(p)(n=1) e(alpha(k) n(k) + alpha(1)n),involving lower order main terms. As an application, we show that for almost all alpha(2) is an element of [0, 1) one hassup(alpha 1 is an element of[0,1)) | Sigma(1 \u3c= n \u3c= P) e(alpha(1)(n(3) + n) + alpha(2)n(3))| \u3c\u3c P3/4+epsilon,and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrodinger and Airy equations

On generating functions in additive number theory, II: Lower-order terms and applications to PDEs

We obtain asymptotics for sums of the form $$\sum_{n=1}^P e(\alpha_kn^k + \alpha_1n),$$ involving lower order main terms. As an application, we show that for almost all $\alpha_2 \in [0,1)$ one has $$\sup_{\alpha_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(\alpha_1(n^3+n) + \alpha_2 n^3) \Big| \ll P^{3/4 + \varepsilon},$$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schr\"odinger and Airy equations