12 problems

Let $R$ be a large parameter and define the thin annulus of lattice points … Assume this annulus contains approximately $R^{1/2}$ lattice points. Thin-annulus lattice-point restric…

For a finite subset of the integer paraboloid, let $\Lambda(r)$ denote the property that its exponential sums satisfy a uniform $L^r$ bound by the associated $\ell^2$ norm. Likewis…

Wang's conjecture. The range of $\beta$ in Proposition can be extended to

Let $p\geq 2$ and $d\geq 2$. Let $\mathbb{T}^d=\mathbb{R}^d/\mathbb{Z}^d$ be the square torus, and let $e_N$ be an eigenfunction of $\sqrt{-\Delta}$ with eigenvalue $2\pi N$. The d…

Let $d$ and $N$ be positive integers, and consider the discrete sphere … A set $E\subset\mathbb{Z}^d$ is a $\Lambda(p)$-set if it has the $\Lambda(p)$-property, namely the relevant…

Let $d\geq 3$ be a natural number, let $m$ be a natural number, let $(\mathfrak{a}_{\bm{n}})_{\bm{n}\in S_{d,m}}$ be complex numbers, let $\epsilon>0$, and let…

Let $\phi\in\mathbb{Z}[n]$ be a polynomial of degree $k\geq3$, and let $a\in\ell^2(\mathbb{Z})$. Define the extension operator … where $\alpha\in\mathbb{T}$ and $e(t)=e^{2\pi i t}$…

Let $\phi_1,\phi_2\in\mathbb{Z}[n]$ be polynomials with respective degrees $k_1$ and $k_2$, and define … where $a\in\ell^2(\mathbb{Z})$ and $(\alpha_1,\alpha_2)\in\mathbb{T}^2$. As…

Let $N\in\mathbb{N}$ and let $a\in\ell^2(\mathbb{Z})$. Define the truncated extension operator … where $(\alpha,\beta)\in\mathbb{T}^2$ and $e(t)=e^{2\pi i t}$. Discrete restriction…

Let $n \geq 3$, let $\lambda \geq 1$, and let $S^{n-1}$ denote the unit sphere in $\mathbb{R}^n$. For a function $f$ on $\mathbb{T}^n$, write $\widehat f(k)$ for its Fourier coeffi…

For $n\ge 3$, $N\ge 1$, and $\lambda=N^2$, define the discrete sphere … For coefficients $a_\xi\in\mathbb C$, write $e(\xi\cdot x)$ for the exponential character on $\mathbb T^n$.…

Let $n\ge 2$, let $P^{n-1}(N)=\{\xi=(\xi_1,\ldots,\xi_n)\in\mathbb Z^n:\xi_n=\xi_1^2+\cdots+\xi_{n-1}^2,\ |\xi_1|,\ldots,|\xi_{n-1}|\le N\}$, and let $a_\xi\in\mathbb C$. The notat…