T. Wooley

activity

20032026

most citedThe cubic case of the main conjecture in Vinogradov's mean value theorem

144 citations · 741 across the 67 of their papers we have counts for

collaborators

72 papers

math.NT2026

Sharp mean value estimates in the absence of translation-dilation invariance

Trevor D. Wooley

We establish sharp mean value estimates for various exponential sums lacking translation-dilation invariance properties. For example, suppose that $k \geq 1$ and $s > 0$ . Then, for eac…

math.NT2026

The main conjecture for a Vinogradov subsystem

Trevor D. Wooley

When $k \geq 2$ and $s > 0$ , we establish that for each $ε > 0$ , one has \[ \int_{[0,1)^{k-1}}\biggl| \sum_{1\le x\le X}e(α_2x^2+\ldots +α_k x^k)\biggr|^{2s}\,{\rm d}\boldsymbo…

math.NT2026

Structure and paucity in affine diagonal systems, I

Julia Brandes, Trevor D. Wooley

Let $ε > 0$ and $h \in Z^{3}$ . 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 $P^{ε}$ …

math.NT2026

Strong paucity in the Brüdern-Robert Diophantine system

Trevor D. Wooley

Let $k$ be a natural number with $k \geq 2$ , and let $ε > 0$ . We consider the number $V_{k}^{*} (P)$ of integral solutions of the system of simultaneous Diophantine equations \[ x…

math.NT2025

Subconvex $L^{p}$ -sets, Weyl's inequality, and equidistribution

Trevor D. Wooley

We examine sets $A$ of natural numbers having the property that for some real number $p \in (0, 2)$ , one has the subconvex bound $$\int_0^1 \Bigl| \sum_{n\in \mathscr A\cap…

math.NT2024

Estimates for smooth Weyl sums on minor arcs

Joerg Bruedern, Trevor D. Wooley

We provide new estimates for smooth Weyl sums on minor arcs and explore their consequences for the distribution of the fractional parts of $α n^{k}$ . In particular, when $k \geq 6$ and…

math.NT2024

Estimates for smooth Weyl sums on major arcs

Joerg Bruedern, Trevor D. Wooley

We present estimates for smooth Weyl sums of use on sets of major arcs in applications of the Hardy-Littlewood method. In particular, we derive mean value estimates on major arcs f…

math.NT2024

Partitio Numerorum: sums of squares and higher powers

Joerg Bruedern, Trevor D. Wooley

We survey the potential for progress in additive number theory arising from recent advances concerning major arc bounds associated with mean value estimates for smooth Weyl sums. W…

math.NT2023

On $2$ -superirreducible polynomials over finite fields

Jonathan W. Bober, Lara Du, Dan Fretwell +2

We investigate $k$ -superirreducible polynomials, by which we mean irreducible polynomials that remain irreducible under any polynomial substitution of positive degree at most $k$ .…

math.NT2023

Condensation and densification for sets of large diameter

Trevor D. Wooley

Consider a set of integers $A$ having finite diameter $X$ , and a system of simultaneous polynomial equations to be solved over $A$ . In many circumstances, it is k…

math.NT2023

Rational lines on diagonal hypersurfaces and subconvexity via the circle method

Trevor D. Wooley

Fix $k, s, n \in N$ , and consider non-zero integers $c_{1}, \dots, c_{s}$ , not all of the same sign. Provided that $s \geq k (k + 1)$ , we establish a Hasse principle for the existence…

math.NT2023★ 3 cited

On Waring's problem: beyond Freiman's theorem

Joerg Bruedern, Trevor D. Wooley

Let $k_{i} \in N$ $(i \geq 1)$ satisfy $2 \leq k_{1} \leq k_{2} \leq \dots$ . Freiman's theorem shows that when $j \in N$ , there exists $s = s (j) \in N$ such that all large i…

math.NT2022

Finite abelian groups via congruences

Trevor D. Wooley

For every finite abelian group $G$ , there are positive integers $n$ and $d$ such that $G$ is isomorphic to the multiplicative group of $d$ -th powers of reduced residues modulo $n$ .

math.NT2022

The paucity problem for certain symmetric Diophantine equations

Trevor D. Wooley

Let $φ_{1}, \dots, φ_{r} \in Z [z_{1}, \dots z_{k}]$ be integral linear combinations of elementary symmetric polynomials with $deg (φ_{j}) = k_{j}$ $(1 \leq j \leq r)$ , where $1\le k_1<k…

math.NT2022

Pairs of diagonal quartic forms: the asymptotic formulae

Joerg Bruedern, Trevor D. Wooley

We establish an asymptotic formula for the number of integral solutions of bounded height for pairs of diagonal quartic equations in $26$ or more variables. In certain cases, pairs…

math.NT2022

Partitio Numerorum: sums of a prime and a number of $k$ -th powers

Joerg Bruedern, Trevor D. Wooley

Let $k$ be a natural number and let $c = 2.134693 \dots$ be the unique real solution of the equation $2 c = 2 + lo g (5 c - 1)$ in $[1, \infty)$ . Then, when $s \geq c k + 4$ , we establish an asymp…

math.NT2022

On Waring's problem for larger powers

Joerg Bruedern, Trevor D. Wooley

Let $G (k)$ denote the least number $s$ having the property that every sufficiently large natural number is the sum of at most $s$ positive integral $k$ -th powers. Then for all $k\i…

math.NT2022★ 1 cited

Subconvexity in inhomogeneous Vinogradov systems

Trevor D. Wooley

When $k$ and $s$ are natural numbers and $h \in Z^{k}$ , denote by $J_{s, k} (X; h)$ the number of integral solutions of the system \[ \sum_{i=1}^s(x_i^j-y_i^j)=h_…

math.NT2022★ 1 cited

Subconvexity in the inhomogeneous cubic Vinogradov system

Trevor D. Wooley

When $h \in Z^{3}$ , denote by $B (X; h)$ the number of integral solutions to the system \[ \sum_{i=1}^6(x_i^j-y_i^j)=h_j\quad (1\le j\le 3), \] with $1\le x_i,y_…

math.NT2022

Subconvexity and the Hilbert-Kamke Problem

Trevor D. Wooley

When $s \geq k \geq 3$ and $n_{1}, \dots, n_{k}$ are large natural numbers, denote by $A_{s, k} (n)$ the number of solutions in non-negative integers $x$ to the system \[ x_1…

math.NT2021

Pairs of diagonal quartic forms: the non-singular Hasse principle

Joerg Bruedern, Trevor D. Wooley

We establish the non-singular Hasse Principle for pairs of diagonal quartic equations in $22$ or more variables.

math.NT2021

On smooth Weyl sums over biquadrates and Waring's problem

Joerg Bruedern, Trevor D. Wooley

We provide estimates for $s^{th}$ moments of biquadratic smooth Weyl sums, when $10 \leq s \leq 12$ , by enhancing the second author's iterative method that delivers estimates beyon…

math.NT2021

A paucity problem associated with a shifted integer analogue of the divisor function

Winston Heap, Anurag Sahay, Trevor D. Wooley

Suppose that $θ$ is irrational. Then almost all elements $ν \in Z [θ]$ that may be written as a $k$ -fold product of the shifted integers $n + θ$ $(n \in N)$ are thus…

math.NT2021★ 2 cited

Paucity problems and some relatives of Vinogradov's mean value theorem

Trevor D. Wooley

When $k \geq 4$ and $0 \leq d \leq (k - 2) /4$ , we consider the system of Diophantine equations \[ x_1^j+\ldots +x_k^j=y_1^j+\ldots +y_k^j\quad (1\le j\le k,\, j\ne k-d). \] We show that in…

Sharp mean value estimates in the absence of translation-dilation invariance

The main conjecture for a Vinogradov subsystem

Structure and paucity in affine diagonal systems, I

Strong paucity in the Brüdern-Robert Diophantine system

Subconvex Lp-sets, Weyl's inequality, and equidistribution

Estimates for smooth Weyl sums on minor arcs

Estimates for smooth Weyl sums on major arcs

Partitio Numerorum: sums of squares and higher powers

On 2-superirreducible polynomials over finite fields

Condensation and densification for sets of large diameter

Rational lines on diagonal hypersurfaces and subconvexity via the circle method

On Waring's problem: beyond Freiman's theorem

Finite abelian groups via congruences

The paucity problem for certain symmetric Diophantine equations

Pairs of diagonal quartic forms: the asymptotic formulae

Partitio Numerorum: sums of a prime and a number of k-th powers

On Waring's problem for larger powers

Subconvexity in inhomogeneous Vinogradov systems

Subconvexity in the inhomogeneous cubic Vinogradov system

Subconvexity and the Hilbert-Kamke Problem

Pairs of diagonal quartic forms: the non-singular Hasse principle

On smooth Weyl sums over biquadrates and Waring's problem

A paucity problem associated with a shifted integer analogue of the divisor function

Paucity problems and some relatives of Vinogradov's mean value theorem

Subconvex $L^{p}$ -sets, Weyl's inequality, and equidistribution

On $2$ -superirreducible polynomials over finite fields

Partitio Numerorum: sums of a prime and a number of $k$ -th powers