arXiv subjects
Publications and source records attributed to Trevor D. Wooley.
Let $φ_1,\ldots ,φ_r\in \mathbb Z[z_1,\ldots z_k]$ be integral linear combinations of elementary symmetric polynomials with $\text{deg}(φ_j)=k_j$ $(1\le j\le r)$, where $1\le k_1<k_2<\ldots <k_r=k$. Subject to the condition $k_1+\ldots +k_r\ge \tfrac{1}{2}k(k-1)+2$, we show that there is a paucity of non-diagonal solutions to the Diophantine system $φ_j(\mathbf x)=φ_j(\mathbf y)$ $(1\le j\le r)$.
When $k$ and $s$ are natural numbers and $\mathbf h\in \mathbb Z^k$, denote by $J_{s,k}(X;\mathbf h)$ the number of integral solutions of the system \[ \sum_{i=1}^s(x_i^j-y_i^j)=h_j\quad (1\le j\le k), \] with $1\le x_i,y_i\le X$. When $s<k(k+1)/2$ and $(h_1,\ldots ,h_{k-1})\ne {\mathbf 0}$, Brandes and Hughes have shown that $J_{s,k}(X;\mathbf h)=o(X^s)$. In this paper we improve on quantitative aspects of this result, and, subject to an extension of the main conjecture in Vinogradov's mean value theorem, we obtain an asymptotic formula for $J_{s,k}(X;\mathbf h)$ in the critical case $s=k(k+1)/2$. The latter requires minor arc estimates going beyond square-root cancellation.
When $\mathbf h\in \mathbb Z^3$, denote by $B(X;\mathbf h)$ 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 $1\le x_i,y_i\le X$ $(1\le i\le 6)$. When $h_1\ne 0$ and appropriate local solubility conditions on $\mathbf h$ are met, we obtain an asymptotic formula for $B(X;\mathbf h)$, thereby establishing a subconvex local-global principle in the inhomogeneous cubic Vinogradov system. We obtain similar conclusions also when $h_1=0$, $h_2\ne 0$ and $X$ is sufficiently large in terms of $h_2$. Our arguments involve minor arc estimates going beyond square-root cancellation.
When $s\ge k\ge 3$ and $n_1,\ldots ,n_k$ are large natural numbers, denote by $A_{s,k}(\mathbf n)$ the number of solutions in non-negative integers $\mathbf x$ to the system \[ x_1^j+\ldots +x_s^j=n_j\quad (1\le j\le k). \] Under appropriate local solubility conditions on $\mathbf n$, we obtain an asymptotic formula for $A_{s,k}(\mathbf n)$ when $s\ge k(k+1)$. This establishes a local-global principle in the Hilbert-Kamke problem at the convexity barrier. Our arguments involve minor arc estimates going beyond square-root cancellation.
We provide estimates for $s^{\rm th}$ moments of biquadratic smooth Weyl sums, when $10\le s\le 12$, by enhancing the second author's iterative method that delivers estimates beyond the classical convexity barrier. As a consequence, all sufficiently large integers $n$ satisfying $n\equiv r\pmod{16}$, with $1\le r\le 12$, can be written as a sum of $12$ biquadrates of smooth numbers.
We establish the non-singular Hasse Principle for pairs of diagonal quartic equations in $22$ or more variables.
Suppose that $θ$ is irrational. Then almost all elements $ν\in {\mathbb Z}[θ]$ that may be written as a $k$-fold product of the shifted integers $n+θ$ $(n\in {\mathbb N})$ are thus represented essentially uniquely.
When $k\ge 4$ and $0\le d\le (k-2)/4$, 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 this cousin of a Vinogradov system, there is a paucity of non-diagonal positive integral solutions. Our quantitative estimates are particularly sharp when $d=o(k^{1/4})$.
We consider certain systems of three linked simultaneous diagonal equations in ten variables with total degree exceeding five. By means of a complification argument, we obtain an asymptotic formula for the number of integral solutions of this system of bounded height that resolves the associated paucity problem.
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.
Defining the truncated extension operator $E$ for a sequence $a(n)$ with $n \in {\mathbb Z}$ by putting \[ E{a}(α,β):=\sum_{|n|\leq N}a(n) e(αn^3 + βn), \] we obtain the conjectured tenth moment estimate \[ \| E{a} \|_{L^{10}({\mathbb T}^2)}\lesssim_εN^{\frac{1}{10}+ε} \|a\|_{\ell^2({\mathbb Z})}. \] We obtain related conclusions when the curve $(x,x^3)$ is replaced by $(ϕ_1(x), ϕ_2(x))$ for suitably independent polynomials $ϕ_1(x),ϕ_2(x)$ having integer coefficients.
We apply the circle method to obtain an asymptotic formula for the number of integral points on a certain sliced cubic hypersurface related to the Segre cubic. Unusually, the major and minor arc integrals in this application are both positive and of the same order of magnitude.
Given $f\in \mathbb{Z}[t]$ of positive degree, we investigate the existence of auxiliary polynomials $g\in \mathbb{Z}[t]$ for which $f(g(t))$ factors as a product of polynomials of small relative degree. One consequence of this work shows that for any quadratic polynomial $f\in\mathbb{Z}[t]$ and any $ε> 0$, there are infinitely many $n\in\mathbb{N}$ for which the largest prime factor of $f(n)$ is no larger than $n^ε$.
We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when $φ_j\in \mathbb Z[t]$ $(1\le j\le k)$ is a system of polynomials with non-vanishing Wronskian, and $s\le k(k+1)/2$, then for all complex sequences $(\mathfrak a_n)$, and for each $ε>0$, one has \[ \int_{[0,1)^k} \left| \sum_{|n|\le X} {\mathfrak a}_n e(α_1φ_1(n)+\ldots +α_kφ_k(n)) \right|^{2s} {\rm d}{\boldsymbol α} \ll X^ε\left( \sum_{|n|\le X} |{\mathfrak a}_n|^2\right)^s. \] As a special case of this result, we confirm the main conjecture in Vinogradov's mean value theorem for all exponents $k$, recovering the recent conclusions of the author (for $k=3$) and Bourgain, Demeter and Guth (for $k\ge 4$). In contrast with the $l^2$-decoupling method of the latter authors, we make no use of multilinear Kakeya estimates, and thus our methods are of sufficient flexibility to be applicable in algebraic number fields, and in function fields. We outline such extensions.
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 $I_{s,k,1}(X)\ll X^{s+ε}$.
When $k>1$ and $n$ is the product of the smallest $k$ primes, the $(k+1)$-st smallest prime is the least divisor exceeding $1$ of $n^{n^n}-1$. This variant of Euclid's prime generator is discussed with some of its cousins.
We apply recent progress on Vinogradov's mean value theorem to improve bounds for the function $H(k)$ in the Waring-Goldbach problem. We obtain new results for all exponents $k \ge 7$, and in particular establish that for large $k$ one has \[H(k)\le (4k-2)\log k-(2\log 2-1)k-3.\]
Let $G(k)$ denote the least number $s$ such that every sufficiently large natural number is the sum of at most $s$ positive integral $k$th powers. We show that $G(7)\le 31$, $G(8)\le 39$, $G(9)\le 47$, $G(10)\le 55$, $G(11)\le 63$, $G(12)\le 72$, $G(13)\le 81$, $G(14)\le 90$, $G(15)\le 99$, $G(16)\le 108$.