arXiv subjects

Simon L. Rydin Myerson

Publications and source records attributed to Simon L. Rydin Myerson.

8 recordsLinked to original sources

arXiv2026

A two-dimensional delta symbol method and its application to pairs of quadratic forms

We present a two-dimensional delta symbol method that facilitates a version of the Kloosterman refinement of the circle method, addressing a question posed by Heath-Brown. As an application, we establish the asymptotic formula for the number of integral points on a non-singular intersection of two integral quadratic forms with at least $10$ variables. Assuming the Generalized Lindelöf Hypothesis, we reduce the number of variables to $9$ by performing a double Kloosterman refinement. A heuristic argument suggests our two-dimensional delta symbol will typically outperform known expressions of this type by an increasing margin as the number of variables grows.

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arXiv2025

Bounds for spectral projectors on the three-dimensional torus

We study $L^2$ to $L^p$ operator norms of spectral projectors for the Euclidean Laplacian on the torus in the case where the spectral window is narrow. With a window of constant size this is a classical result of Sogge; in the small-window limit we are left with $L^p$ norms of eigenfunctions of the Laplacian, as considered for instance by Bourgain. For the three-dimensional torus we prove new cases of a previous conjecture of the first two authors concerning the size of these norms; we also refine certain prior results to remove $ε$-losses in all dimensions. We use methods from number theory: the geometry of numbers, the circle method and exponential sum bounds due to Guo. We complement these techniques with height splitting and a bilinear argument to prove sharp results. We exposit on the various techniques used and their limitations.

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arXiv2023

The elliptic sieve and Brauer groups

A theorem of Serre states that almost all plane conics over $\mathbb{Q}$ have no rational point. We prove an analogue of this for families of conics parametrised by elliptic curves using elliptic divisibility sequences and a version of the Selberg sieve for elliptic curves. We also give more general results for specialisations of Brauer groups, which yields applications to norm form equations.

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arXiv2023

Additive problems with almost prime squares

We show that every sufficiently large integer is a sum of a prime and two almost prime squares, and also a sum of a smooth number and two almost prime squares. The number of such representations is of the expected order of magnitude. We likewise treat representations of shifted primes p-1 as sums of two almost prime squares. The methods involve a combination of analytic, automorphic and algebraic arguments to handle representations by restricted binary quadratic forms with a high degree of uniformity.

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arXiv2022

Bounds for spectral projectors on generic tori

We investigate norms of spectral projectors on thin spherical shells for the Laplacian on generic tori, including generic rectangular tori. We state a conjecture and partially prove it, improving on previous results concerning arbitrary tori.

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arXiv2022

Quadratic forms and systems of forms in many variables

Let $F_1,\dotsc,F_R$ be quadratic forms with integer coefficients in $n$ variables. When $n\geq 9R$ and the variety $V(F_1,\dotsc,F_R)$ is a smooth complete intersection, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish, which in particular implies the Hasse principle for $V(F_1,\dotsc,F_R)$. Previous work in this direction required $n$ to grow at least quadratically with $R$. We give a similar result for $R$ forms of degree $d$, conditional on an upper bound for the number of solutions to an auxiliary inequality. In principle this result may apply as soon as $n> d2^dR$. In the case that $d\geq 3$, several strategies are available to prove the necessary upper bound for the auxiliary inequality. In a forthcoming paper we use these ideas to apply the circle method to nonsingular systems of forms with real coefficients.

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arXiv2022

Bounds for spectral projectors on the Euclidean cylinder

We prove essentially optimal bounds for norms of spectral projectors on thin spherical shells for the Laplacian on the cylinder (R/Z)*R. In contrast to previous investigations into spectral projectors on tori, having one unbounded dimension available permits a compact self-contained proof.

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arXiv2017

Systems of cubic forms in many variables

We consider a system of $R$ cubic forms in $n$ variables, with integer coefficients, which define a smooth complete intersection in projective space. Provided $n\geq 25R$, we prove an asymptotic formula for the number of integer points in an expanding box at which these forms simultaneously vanish. In particular we can handle systems of forms in $O(R)$ variables, previous work having required that $n \gg R^2$. One conjectures that $n \geq 6R+1$ should be sufficient. We reduce the problem to an upper bound for the number of solutions to a certain auxiliary inequality. To prove this bound we adapt a method of Davenport.

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