33 problems

Consider the problem of finding an optimal lattice sampling pattern for Gaussian Gabor systems. Let a lattice mean a discrete subgroup of the relevant phase space, and compare latt…

Gramain's conjecture.

Uniqueness conjecture. There are no other extreme Delaunay polytopes in dimension $7$.

Let $A_n(D)$ be the maximum cardinality of a subset $\mathcal F\subseteq\mathbb Z^n$ whose diameter in the anticode metric $d_{\mathrm a}$ is at most $D$. Let $S_{n,D}$ be the simp…

Let a space-filling polyhedron be a convex polyhedron that tiles three-dimensional space by translations, and restrict to such polyhedra of unit volume. The truncated octahedron is…

Let $b/a=[a_1,a_2,\dots,a_n]$ be the continued-fraction expansion of the rational number $b/a$, and let $A_m,B_m$ be the corresponding penultimate-continuant vertices of the Markov…

Markov Sail Duality conjecture. The coefficients of Markov polynomials satisfy these two Markov-sail duality identities. This would determine almost all interior sail coefficients…

Let $\rho=a/b$ be rational, and consider the Markov sail formed by the integer lattice points in the critical triangle of the Newton polygon, together with the coefficients of the…

Lattice-minimality conjecture. This lattice spans at least

For a convex body $\operatorname{\mathcal{P}} \subseteq \operatorname{\mathbb{R}}^n$, its lattice-width is … If…

Average-distance conjecture. For Euclidean distances, the average distance from a uniformly distributed point in $\mathbb{R}^D$ to the nearest point of $\mathcal{L}$ is not less th…

Restrict $\Gamma$ to be a rectangular lattice, and let $\mathcal{L}_{\mathbb{Z}^2}$ denote the corresponding Landau constant. Rectangular-lattice conjecture. For the problem restri…

Let $\Gamma \subset \mathbb{C}$ be discrete and $f_\Gamma \in \mathcal{U}$, where $\mathcal{U}$ is the class of universal covering maps of $\mathbb{D}$ onto…

Critical-point conjecture. The function $\mathcal{J}(z;a,b)$ has either four or six critical points with respect to $a,b$, and the following hold:

Let $D>0$ be an exceptional distance, and let $\mathcal{T}(D)$ be the set of acute-angled triangles inscribed in $\mathbb{Z}^2$ with all side lengths at least $D$. Sliding occurs w…

Let $mathcal{S}subsetmathbb{Z}^2$ be a finite convex lattice set, meaning that its convex hull $operatorname{Conv}(mathcal{S})subsetmathbb{R}^2$ is closed and … For a configurati…

Strong dimension conjecture. For every dual $k$-cell $D$ there is a $k$-dimensional lattice $\Lambda$ such that there is a $k$-dimensional cell in the Delone tessellation of…

Let $S=\operatorname{conv}(\{v_0,\ldots,v_d\})$ be a $d$-simplex with $\mathbf{0}\in S\setminus\{v_0,\ldots,v_d\}$ and with rational vertex directions. For each $i$, let…

Let $S$ be a $d$-simplex with the origin in its interior and with rational vertex directions. Let $\operatorname{Vol}_{\mathbb{Z}^d}(S)$ be normalized volume and let … where…

Let $r$ and $r^{\ast}$ be positive continuously differentiable functions on the circle, and let $Z_r(s)$ denote the Hlawka zeta function associated with $r$. The group…

Let $r\bigl(\theta\bigr)$ be a positive continuous function on the circle, viewed as a radial function, and suppose that infinitely many Fourier coefficients $\hat{r}(n)$ are nonze…

Let $n$ be a positive integer. Let $Z\subseteq\mathbb{R}^n$ be a zonotope generated by $n+1$ vectors of $\mathbb{Z}^n$ in LGP (general linear position), and let $\boldsymbol{x}$ be…

Let $S=\{\boldsymbol{z}_1,\dotsc,\boldsymbol{z}_m\}\subseteq\mathbb{Z}^n$ and let $Z$ be the lattice zonotope generated by $S$. The set $S$ is in LGP (general linear position) when…

Let $G$ be the lighter degree-$3$ spanner of the infinite square lattice shown in Fig., and let $\delta_0$ denote the minimum stretch factor for degree-$3$ spanners of the lattice.…

Let $\Lambda$ be the infinite square lattice, and let $\delta_0(\Lambda,3)$ denote its minimum stretch factor among degree-$3$ spanners. Degree 3 dilation conjecture. … The paper h…