9 problems

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

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

Condensation and densification conjecture. One has

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

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…

The Main Conjecture. For all integers $s,k\geq 1$,

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

Let $J_{r,d}(X)$ denote the number of solutions in integers $1\leq x_1,\ldots,x_{2r}\leq X$ to … where $r,d\geq 1$, and set $D=d(d+1)/2$. Vinogradov's main conjecture. For every…