On Waring's problem: some consequences of Golubeva's method

We investigate sums of mixed powers involving two squares, two cubes, and various higher powers, concentrating on situations inaccessible to the Hardy-Littlewood method

Mean value estimates for odd cubic Weyl sums

We establish an essentially optimal estimate for the ninth moment of the exponential sum having argument $\alpha x^3+\beta x$. 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

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 $1\le s\le \frac{1}{2}k(k+1)$.Comment: arXiv admin note: text overlap with arXiv:1310.844

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 $\tfrac{5}{8}k^2<s\le k^2-k+1$. There are numerous applications. In particular, when $k$ is large, the anticipated asymptotic formula in Waring's problem is established for sums of $s$ $k$th powers of natural numbers whenever $s>1.543k^2$. The asymptotic formula is also established for sums of $28$ fifth powers.Comment: 48pp; modest revisions in light of referee comment

On Waring's problem: two squares, two cubes and two sixth powers

We investigate the number of representations of a large positive integer as the sum of two squares, two positive integral cubes, and two sixth powers, showing that the anticipated asymptotic formula fails for at most O((log X)^3) positive integers not exceeding X.Comment: Corrected quotient of gamma functions in singular integra

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 $$\int_{[0,1)^k}\left| \sum_{|n|\le N}\mathfrak a_ne(\alpha_1n+\ldots +\alpha_kn^k) \right|^{2s}\,{\rm d}{\mathbf \alpha}\ll N^{s-k(k+1)/2}\biggl( \sum_{|n|\le N}|\mathfrak a_n|^2\biggr)^s.$$ 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.Comment: 37 page

On Waring's problem for intermediate powers

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 $G(7)\le 31$, $G(8)\le 39$, $G(9)\le 47$, $G(10)\le 55$, $G(11)\le 63$, $G(12)\le 72$, $G(13)\le 81$, $G(14)\le 90$, $G(15)\le 99$, $G(16)\le 108$.Comment: 7 page

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 $\Bbb Q$.Comment: 11 page

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.Comment: Submitted to Proceedings of the ICM 201