18 problems

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

Let $k\geq 2$ be a fixed integer and let $\epsilon>0$. Let … where the leading coefficient satisfies $|\alpha_k-a/q|\leq 1/q^2$ for coprime integers $a,q$. Write…

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…

Maximal Weyl-sum conjecture. The maximal estimate

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

Endpoint fourth-moment conjecture. The conjecture is

Let $2<r<4$, let $N$ be a positive integer, and let $a_1,\ldots,a_N$ be complex numbers. Write $\mathsf{T}=\mathbb{R}/\mathbb{Z}$ and $\mathbf{e}(t)=e^{2\pi i t}$. Rudin's conjectu…

For $d\geqslant 3$, let $\mathsf{T}_d$ denote the $d$-dimensional torus and let … be the corresponding Weyl sum. Optimal-exponent conjecture. For almost all…

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…

Forrest's conjecture. Theorem is true for every irrational $x$ satisfying

The almost-everywhere Weyl-sum exponent conjecture. For each integer $d\geqslant 2$ we have

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 $\tilde{G}(k)$ be the smallest integer $s$ for which the circle-method asymptotic formula for the number $r_{s,k}(l)$ of representations of $l$ as a sum of $s$ positive $k$th p…

Let … For $s\geq k\geq3$, elementary estimates give the lower bound $\mathbf{I}_{s,k}(X)\gg X^{2s-k}$. Optimal mean-value conjecture. In this range, $\mathbf{I}_{s,k}(X)$ has true…

Paucity conjecture. If $2\leq s<k$, the diagonal solutions dominate, so that

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 $\epsilon>0$,…