arXiv subjects

Julia Brandes

Publications and source records attributed to Julia Brandes.

At least 19 recordsLinked to original sources

arXiv2026

Structure and paucity in affine diagonal systems, I

Let $\varepsilon>0$ and $\mathbf h\in \mathbb Z^3$. We show that whenever $P$ is large and the system \[ x_1^j+x_2^j-y_1^j-y_2^j=h_j\quad (j=1,2,3) \] has more than $P^\varepsilon$ integral solutions with $1\le x_i,y_i\le P$, then there exist natural numbers $a$ and $b$ with $h_j=a^j-b^j$ $(j=1,2,3)$. This example illustrates the theme that, either the Diophantine system has a paucity of integral solutions, or else the coefficient tuple $\mathbf h$ is highly structured. We examine related paucity problems as well as some consequences for problems involving more variables.

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arXiv2023

Rational lines on cubic hypersurfaces II

We show that any rational cubic hypersurface of dimension at least 33 defined over a number field $K$ vanishes on a $K$-rational projective line, reducing the previous lower bound of Wooley by two. For $K=\mathbb Q$ we can reduce the bound to 29. The main ingredients are a result on linear spaces on quadratic forms over suitable non-real quadratic field extensions, and recent work of Bernert and Hochfilzer on cubic forms over imaginary quadratic number fields for the rational case.

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arXiv2023

Campana points on diagonal hypersurfaces

We construct an integral model for counting Campana points of bounded height on diagonal hypersurfaces of degree greater than one, and give an asymptotic formula for their number, generalising work by Browning and Yamagishi. The paper also includes background material on the theory of Campana points on hyperplanes and previous results in the field.

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arXiv2023

A minimalist version of the circle method and Diophantine problems over thin sets

This paper studies the minimal conditions under which we can establish asymptotic formulae for Waring's problem and other additive problems that may be tackled by the circle method. We confirm in quantitative terms the well-known heuristic that a mean value estimate and an estimate of Weyl type, together with suitable distribution properties of the underlying set over a set of admissible residue classes, are sufficient to implement the circle method. This allows us to give a rather general proof of Waring's problem which is applicable to a range of sufficiently well-behaved thin sets, such as the ellipsephic sets recently investigated by the first author.

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arXiv2023

Local mean value estimates for Weyl sums

We obtain new estimates - both upper and lower bounds - on the mean values of the Weyl sums over a small box inside of the unit torus. In particular, we refine recent conjectures of C. Demeter and B. Langowski (2022), and improve some of their results.

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arXiv2022

Reinforcing a Philosophy: A counting approach to square functions over local fields

In this paper, we study square functions for extension operators over finite-type, planar curves endowed with the Euclidean arclength measure. We prove new results for curves of the form $(T,\phi(T))$ where $\phi(T)$ is a polynomial of degree at least 2. This includes new estimates for such curves given by monomials $\phi(T) = T^k$ for $k \geq 3$ which are uniform over all local fields whose characteristic is coprime to $k$. Key to our approach is a systematic analysis of the second order differencing polynomial and its geometry in local fields.

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arXiv2021

On the inhomogeneous Vinogradov system

We show that the system of equations \begin{align*} \sum_{i=1}^s (x_i^j-y_i^j) = a_j \qquad (1 \le j \le k) \end{align*} has appreciably fewer solutions in the subcritical range $s < k(k+1)/2$ than its homogeneous counterpart, provided that $a_\ell \neq 0$ for some $\ell \le k-1$. Our methods use Vinogradov's mean value theorem in combination with a shifting argument.

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arXiv2020

Two-dimensional Weyl sums failing square-root cancellation along lines

We show that a certain two-dimensional family of Weyl sums of length $P$ takes values as large as $P^{3/4 + o(1)}$ on almost all linear slices of the unit torus, contradicting a widely held expectation that Weyl sums should exhibit square-root cancellation on generic subvarieties of the unit torus. This is an extension of a result of J. Brandes, S. T. Parsell, C. Poulias, G. Shakan and R. C. Vaughan (2020) from quadratic and cubic monomials to general polynomials of arbitrary degree. The new ingredients of our approach are the classical results of E. Bombieri (1966) on exponential sums along a curve and R. J. Duffin and A. C. Schaeffer (1941) on Diophantine approximations by rational numbers with prime denominators.

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arXiv2020

Optimal mean value estimates beyond Vinogradov's mean value theorem

We establish improved mean value estimates associated with the number of integer solutions of certain systems of diagonal equations, in some instances attaining the sharpest conjectured conclusions. This is the first occasion on which bounds of this quality have been attained for Diophantine systems not of Vinogradov type. As a consequence of this progress, whenever $u \ge 3v$ we obtain the Hasse principle for systems consisting of $v$ cubic and $u$ quadratic diagonal equations in $6v+4u+1$ variables, thus attaining the convexity barrier for this problem.

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arXiv2020

The density of rational lines on hypersurfaces: A bihomogeneous perspective

Let $F$ be a non-singular homogeneous polynomial of degree $d$ in $n$ variables. We give an asymptotic formula of the pairs of integer points $(\mathbf x, \mathbf y)$ with $|\mathbf x| \le X$ and $|\mathbf y| \le Y$ which generate a line lying in the hypersurface defined by $F$, provided that $n > 2^{d-1}d^4(d+1)(d+2)$. In particular, by restricting to Zariski-open subsets we are able to avoid imposing any conditions on the relative sizes of $X$ and $Y$.

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arXiv2020

Rational lines on cubic hypersurfaces

We show that any smooth projective cubic hypersurface of dimension at least $29$ over the rationals contains a rational line. A variation of our methods provides a similar result over p-adic fields. In both cases, we improve on previous results due to the second author and Wooley. We include an appendix in which we highlight some slight modifications to a recent result of Papanikolopoulos and Siksek. It follows that the set of rational points on smooth projective cubic hypersurfaces of dimension at least 29 is generated via secant and tangent constructions from just a single point.

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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.

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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.

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arXiv2017

On the number of linear spaces on hypersurfaces with a prescribed discriminant

For a given form $F\in \mathbb Z[x_1,\dots,x_s]$ we apply the circle method in order to give an asymptotic estimate of the number of $m$-tuples $\mathbf x_1, \dots, \mathbf x_m$ spanning a linear space on the hypersurface $F(\mathbf x) = 0$ with the property that $\det ( (\mathbf x_1, \dots, \mathbf x_m)^t \, (\mathbf x_1, \dots, \mathbf x_m)) = b$. This allows us in some measure to count rational linear spaces on hypersurfaces whose underlying integer lattice is primitive.

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arXiv2017

Vinogradov systems with a slice off

Let $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations $$x_1^j+\ldots +x_s^j=y_1^j+\ldots +y_s^j\quad (\text{$1\le j\le k$, $j\ne r$}),$$ with $1\le x_i,y_i\le X$ $(1\le i\le s)$. By exploiting sharp estimates for an auxiliary mean value, we obtain bounds for $I_{s,k,r}(X)$ for $1\le r\le k-1$. In particular, when $s,k\in \mathbb N$ satisfy $k\ge 3$ and $1\le s\le (k^2-1)/2$, we establish the essentially diagonal behaviour $I_{s,k,1}(X)\ll X^{s+ε}$.

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arXiv2016

The Hasse principle for systems of quadratic and cubic diagonal equations

Employing Brüdern's and Wooley's new complification method, we establish an asymptotic Hasse principle for the number of solutions to a system of r_3 cubic and r_2 quadratic diagonal forms, when the number of cubic equations is at least double the number of quadratic forms. The number of variables required is lower than in earlier work. In particular, we achieve essentially square root cancellation for systems of one quadratic and arbitrarily many cubic equations.

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arXiv2016

Linear spaces on hypersurfaces over number fields

We establish an analytic Hasse principle for linear spaces of affine dimension m on a complete intersection over an algebraic field extension K of Q. The number of variables required to do this is no larger than what is known for the analogous problem over Q. As an application we show that any smooth hypersurface over K whose dimension is large enough in terms of the degree is K-unirational, provided that either the degree is odd or K is totally imaginary.

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arXiv2015

Forms representing forms: The definite case

Let $ψ$ and $F$ be positive definite forms with integral coefficients of equal degree. Using the circle method, we establish an asymptotic formula for the number of identical representations of $ψ$ by $F$, provided $ψ$ is everywhere locally representable and the number of variables of $F$ is large enough. In the quadratic case this supersedes a recent result due to Dietmann and Harvey. Another application addresses the number of primitive linear spaces contained in a hypersurface.

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