Related papers: Approximating the main conjecture in Vinogradov's …

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…

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 $1\le s\le \tfrac{1}{4}(k+1)^2$,…

We apply the efficient congruencing method to estimate Vinogradov's integral for moments of order 2s, with 1<=s<=k^2-1. Thereby, we show that quasi-diagonal behaviour holds when s=o(k^2), we obtain near-optimal estimates for…

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…

We apply a nested variant of multigrade efficient congruencing to estimate mean values related to that of Vinogradov. We show that when $\varphi_j\in \mathbb Z[t]$ $(1\le j\le k)$ is a system of polynomials with non-vanishing Wronskian, and…

We obtain estimates for Vinogradov's integral which for the first time approach those conjectured to be the best possible. Several applications of these new bounds are provided. In particular, the conjectured asymptotic formula in Waring's…

We prove the main conjecture in Vinogradov's Mean Value Theorem for degrees higher than three. This will be a consequence of a sharp decoupling inequality for curves

This is an expository paper, giving a simplified proof of the cubic case of the main conjecture for Vinogradov's mean value theorem.

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…

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…

We obtain sharp estimates for multidimensional generalisations of Vinogradov's mean value theorem for arbitrary translation-dilation invariant systems, achieving constraints on the number of variables approaching those conjectured to be the…

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…

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…

We give a slight refinement to the process by which estimates for exponential sums are extracted from bounds for Vinogradov's mean value. Coupling this with the recent works of Wooley, and of Bourgain, Demeter and Guth, providing optimal…

Given a family $\varphi= (\varphi_1, \ldots, \varphi_d)\in \mathbb{Z}[T]^d$ of $d$ distinct nonconstant polynomials, a positive integer $k\le d$ and a real positive parameter $\rho$, we consider the mean value $$ M_{k, \rho} (\varphi, N) =…

We combine Wooley's efficient congruencing method with earlier work of Vinogradov and Hua to get effective bounds on Vinogradov's mean value theorem.

Let $p$ be a prime, let $s \geq 3$ be a natural number and let $A \subseteq \mathbb{F}_p$ be a non-empty set satisfying $|A| \ll p^{1/2}$. Denoting $J_s(A)$ to be the number of solutions to the system of equations \[ \sum_{i=1}^{s} (x_i -…

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…

In this paper, we provide novel mean value estimates for exponential sums related to the extended main conjecture of Vinogradov's mean value theorem, by developing the Hardy-Littlewood circle method together with a refined shifting…

We establish that three well-known and rather different looking conjectures about Dirichlet characters and their (weighted) sums, (concerning the P\'{o}lya-Vinogradov theorem for maximal character sums, the maximal admissible range in…