arXiv subjects

Scott T. Parsell

Publications and source records attributed to Scott T. Parsell.

5 recordsLinked to original sources

arXiv2020

The Hasse principle for diagonal forms restricted to lower-degree hypersurfaces

We establish the analytic Hasse principle for Diophantine systems consisting of one diagonal form of degree $k$ and one general form of degree $d$, where $d$ is smaller than $k$. By employing a hybrid method that combines ideas from the study of general forms with techniques adapted to the diagonal case, we are able to obtain bounds that grow exponentially in $d$ but only quadratically in $k$, reflecting the growth rates typically obtained for both problems separately. We also discuss some of the most interesting generalisations of our approach.

math.NT↗

arXiv2020

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

We obtain asymptotics for sums of the form $$ \sum_{n=1}^P e(α_kn^k + α_1n), $$ involving lower order main terms. As an application, we show that for almost all $α_2 \in [0,1)$ one has $$ \sup_{α_1 \in [0,1)} \Big| \sum_{1 \le n \le P} e(α_1(n^3+n) + α_2 n^3) \Big| \ll P^{3/4 + \varepsilon}, $$ and that in a suitable sense this is best possible. This allows us to improve bounds for the fractal dimension of solutions to the Schrödinger and Airy equations.

math.NT↗

arXiv2015

Simultaneous additive equations: Repeated and differing degrees

We obtain bounds for the number of variables required to establish Hasse principles, both for existence of solutions and for asymptotic formulae, for systems of additive equations containing forms of differing degree but also multiple forms of like degree. Apart from the very general estimates of Schmidt and Browning--Heath-Brown, which give weak results when specialized to the diagonal situation, this is the first result on such "hybrid" systems.

math.NT↗

arXiv2012

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.

math.NT↗

arXiv2012

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.

math.NT↗