21 problems

Let $k\geq 2$, let $s$ be a positive integer, and define … where $e(\theta)=\exp(2\pi i\theta)$ and $X\geq 1$. The main conjecture in Vinogradov's mean value theorem. For every…

For a natural number $d$, let $n_0(d)$ be the smallest even natural number such that, for every $\boldsymbol{\epsilon}>0$, … The lower bound $n_0(d)\geq 2d$ is known. Threshold con…

The conjectural Weyl-sum estimate. For $d\geq2$, $0\leq u\leq d-1$, and even $p$, one has

The extended main conjecture. One has

Set-uniform Vinogradov mean value conjecture. For every real number $\epsilon>0$,

Condensation and densification conjecture. One has

Let $d$ be a positive integer, let $\alpha\in[0,d]$, let $s\geqslant0$, let $N$ be sufficiently large, and let $\delta$ satisfy … Let $\mathbf a$ be a coefficient vector and let…

Let $d$, $s$, $N$, and … s geqslant frac14d(d+1)+1 qquad text{or} qquad delta geqslant N^{1/d-(d+1)/4}, … I{s,d}^{sharp}(delta;N)leqslant delta^dN^{s+o(1)}+N^{2s-s(d)+o(1)}. … delt…

The large-set extension. Whenever $s>0$ and

The extended main conjecture. Suppose that $k\in\mathbb N$ and ${\mathfrak B}\subseteq[0,1)^k$ is measurable. Whenever $s\in\mathbb R$ satisfies $s\geq\tfrac14k(k+1)+1$, one has

Paucity conjecture. When $1\leqslant s<\tfrac{1}{2}k(k+1)$, one should have

Generalized Vinogradov mean value conjecture. For every $\varepsilon>0$,

Let $J_{s,2}(A)$ denote the number of solutions to the quadratic Vinogradov system for a finite set $A$ of real numbers, with $s$ pairs of variables. Let $s\geq 3$ be a natural num…

For $k,t\in\mathbb{N}$, let … where $\boldsymbol{\alpha}=(\alpha_1,\dots,\alpha_k)\in[0,1)^k$. Vinogradov's mean value conjecture. For each $\varepsilon>0$, … This is the main conj…

Main conjecture. When all of the coefficient matrices $C^{(l)}$ are highly non-singular, the number of integral solutions ${\mathbf x},{\mathbf y}\in[-X,X]^s$ should be at most of…

For each integer $s\ge1$, let $J_{s,2,2}(N)$ denote the number of integral solutions with $1\le X_i,Y_i\le N$ to the quadratic Parsell–Vinogradov system consisting of the conservat…

Main Conjecture. For every $\varepsilon>0$ and every complex sequence $({\mathfrak a}_n)$, the displayed bound holds.

Let $k\geqslant 2$, let $X$ be positive, and define … where ${\boldsymbol \alpha}\in\mathbb R^k$ and $e(z)=e^{2\pi iz}$. For a fixed positive number $s$, write…

Let $f$ be a fixed real-valued polynomial of total degree $d$ in $k$ variables. For each multi-index $\beta=(\beta_1,\ldots,\beta_k)\in\mathbb{Z}_{\geq 0}^k$, write…

Small-$s$ asymptotic conjecture. For $1\leq s\leq 5$, one should have

Vinogradov mean value conjecture. For every $\varepsilon>0$, one has