A low-energy decomposition theorem
We prove that any finite set of real numbers can be split into two parts, one part being highly non-additive and the other highly non-multiplicative.
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arXiv subjects
Trevor D. Wooley
Publications and source records attributed to Trevor D. Wooley.
On the Waring--Goldbach problem for eighth and higher powers
Recent progress on Vinogradov's mean value theorem has resulted in improved estimates for exponential sums of Weyl type. We apply these new estimates to obtain sharper bounds for the function $H(k)$ in the Waring--Goldbach problem. We obtain new results for all exponents $k\ge 8$, and in particular establish that $H(k)\le (4k-2)\log k+k-7$ when $k$ is large, giving the first improvement on the classical result of Hua from the 1940s.
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Discrete Fourier restriction via efficient congruencing: basic principles
We show that whenever $s>k(k+1)$, then for any complex sequence $(\mathfrak a_n)_{n\in \mathbb Z}$, one has $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(α_1n+\ldots +α_kn^k) \right|^{2s}\,{\rm d}{\mathbf α}\ll N^{s-k(k+1)/2}\biggl( \sum_{|n|\le N}|\mathfrak a_n|^2\biggr)^s.$$ Bounds for the constant in the associated periodic Strichartz inequality from $L^{2s}$ to $l^2$ of the conjectured order of magnitude follow, and likewise for the constant in the discrete Fourier restriction problem from $l^2$ to $L^{s'}$, where $s'=2s/(2s-1)$. These bounds are obtained by generalising the efficient congruencing method from Vinogradov's mean value theorem to the present setting, introducing tools of wider application into the subject.
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Arithmetic harmonic analysis for smooth quartic Weyl sums: three additive equations
We establish the non-singular Hasse principle for systems of three diagonal quartic equations in 32 or more variables, subject to a certain rank condition. Our methods employ the arithmetic harmonic analysis of smooth quartic Weyl sums and also a new estimate for their tenth moment.
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Multigrade efficient congruencing and Vinogradov's mean value theorem
We develop a multigrade enhancement of the efficient congruencing method to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, thereby obtaining near-optimal estimates for $\tfrac{5}{8}k^2 1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.
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The Hasse principle for systems of diagonal cubic forms
We establish the Hasse Principle for systems of r simultaneous diagonal cubic equations whenever the number of variables exceeds 6r and the associated coefficient matrix contains no singular r x r submatrix, thereby achieving the theoretical limit of the circle method for such systems.
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Correlation estimates for sums of three cubes
We establish estimates for linear correlation sums involving sums of three positive integral cubes. Under appropriate conditions, the underlying methods permit us to establish the solubility of systems of homogeneous linear equations in sums of three positive cubes whenever these systems have more than twice as many variables as equations.
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Additive representation in short intervals, II: sums of two like powers
We establish that, for almost all natural numbers $N$, there is a sum of two positive integral cubes lying in the interval $[N-N^{7/18+ε},N]$. Here, the exponent $7/18$ lies half way between the trivial exponent $4/9$ stemming from the greedy algorithm, and the exponent $1/3$ constrained by the number of integers not exceeding $X$ that can be represented as the sum of two positive integral cubes. We also provide analogous conclusions for sums of two positive integral $k$-th powers when $k\ge 4$.
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Perturbations of Weyl sums
Write $f_k({\boldsymbol α};X)=\sum_{x\le X}e(α_1x+\ldots +α_kx^k)$ $(k\ge 3)$. We show that there is a set ${\mathfrak B}\subseteq [0,1)^{k-2}$ of full measure with the property that whenever $(α_2,\ldots ,α_{k-1})\in {\mathfrak B}$ and $X$ is sufficiently large, then $$\sup_{(α_1,α_k)\in [0,1)^2}|f_k({\boldsymbol α};X)|\le X^{1/2+4/(2k-1)}.$$ For $k\ge 5$, this improves on work of Flaminio and Forni, in which a Diophantine condition is imposed on $α_k$, and the exponent of $X$ is $1-2/(3k(k-1))$.
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Sums of three cubes, II
Estimates are provided for $s$th moments of cubic smooth Weyl sums, when $4\le s\le 8$, by enhancing the author's iterative method that delivers estimates beyond classical convexity. As a consequence, an improved lower bound is presented for the number of integers not exceeding $X$ that are represented as the sum of three cubes of natural numbers.
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Solvable points on smooth projective varieties
We establish that smooth, geometrically integral projective varieties of small degree are not pointless in suitable solvable extensions of their field of definition, provided that this field is algebraic over $\Bbb Q$.
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On Sums of Powers of Almost Equal Primes
We investigate the Waring-Goldbach problem of representing a positive integer $n$ as the sum of $s$ $k$th powers of almost equal prime numbers. Define $s_k=2k(k-1)$ when $k\ge 3$, and put $s_2=6$. In addition, put $θ_2=\frac{19}{24}$, $θ_3=\frac{4}{5}$ and $θ_k=\frac{5}{6}$ $(k\ge 4)$. Suppose that $n$ satisfies the necessary congruence conditions, and put $X=(n/s)^{1/k}$. We show that whenever $s>s_k$ and $\varepsilon>0$, and $n$ is sufficiently large, then $n$ is represented as the sum of $s$ $k$th powers of prime numbers $p$ with $|p-X|\le X^{θ_k+\varepsilon}$. This conclusion is based on a new estimate of Weyl-type specific to exponential sums having variables constrained to short intervals.
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Rational solutions of pairs of diagonal equations, one cubic and one quadratic
We obtain an essentially optimal estimate for the moment of order 32/3 of the exponential sum having argument $αx^3+βx^2$. Subject to modest local solubility hypotheses, we thereby establish that pairs of diagonal Diophantine equations, one cubic and one quadratic, possess non-trivial integral solutions whenever the number of variables exceeds 10.
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Translation invariance, exponential sums, and Waring's problem
We describe mean value estimates for exponential sums of degree exceeding 2 that approach those conjectured to be best possible. The vehicle for this recent progress is the efficient congruencing method, which iteratively exploits the translation invariance of associated systems of Diophantine equations to derive powerful congruence constraints on the underlying variables. There are applications to Weyl sums, the distribution of polynomials modulo 1, and other Diophantine problems such as Waring's problem.
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Mean value estimates for odd cubic Weyl sums
We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument $αx^3+βx$. The first substantial advance in this topic for over 60 years, this leads to improvements in Heath-Brown's variant of Weyl's inequality, and other applications of Diophantine type.
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On Vinogradov's mean value theorem: strongly diagonal behaviour via efficient congruencing
We enhance the efficient congruencing method for estimating Vinogradov's integral for moments of order $2s$, with $1\le s\le k^2-1$. In this way, we prove the main conjecture for such even moments when $1\le s\le \tfrac{1}{4}(k+1)^2$, showing that the moments exhibit strongly diagonal behaviour in this range. There are improvements also for larger values of $s$, these finding application to the asymptotic formula in Waring's problem.
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The cubic case of the main conjecture in Vinogradov's mean value theorem
We apply a variant of the multigrade efficient congruencing method to estimate Vinogradov's integral of degree $3$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le 6$. Consequently, the main conjecture is now known to hold for the first time in a case of degree exceeding $2$.
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Approximating the main conjecture in Vinogradov's mean value theorem
We apply multigrade efficient congruencing to estimate Vinogradov's integral of degree $k$ for moments of order $2s$, establishing strongly diagonal behaviour for $1\le s\le \frac{1}{2}k(k+1)-\frac{1}{3}k+o(k)$. In particular, as $k\rightarrow \infty$, we confirm the main conjecture in Vinogradov's mean value theorem for 100% of the critical interval $1\le s\le \frac{1}{2}k(k+1)$.
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