9 problems

Almost-everywhere Weyl-sum exponent conjecture. For $\alpha\in[0,1]$,

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

Cordoba's conjecture. For $\alpha<2k$, there exists a constant $C_\alpha$ such that

Endpoint fourth-moment conjecture. The conjecture is

Define … Let $\mathcal X=\{(t,t^2):t\in[0,1]\}$ and let $\mu$ be the natural probability measure on $\mathcal X$. Parabolic natural-measure conjecture. … This asserts that the larg…

For an integer $d\geqslant2$ and $\alpha\in(0,1)$, define … Let $\mathcal X=\{(t,t^2):t\in[0,1]\}\subset\mathsf{T}_2$. Parabolic restriction dimension conjecture. … This is a conje…

For an integer $d\geqslant2$ and $\mathbf x\in\mathsf{T}_d$, let … be the orbit of the Weyl sums. Weyl-sum orbit density conjecture. For almost all $\mathbf x\in\mathsf{T}_d$ in th…

Let $k\geq 3$, set … and suppose that $|\alpha_{k}-a/q|<q^{-2}$ for coprime integers $(a,q)=1$. Let $S_{az}(\boldsymbol{\alpha})$ denote the Weyl sum used in the paper. Conjectural…

Let $r_{k,k}(l)$ denote the number of representations of a positive integer $l$ as a sum of $k$ positive $k$th powers. Hooley's Hypothesis $K^*$. For $k=2$ and every $\epsilon>0$,…