13 problems

Three-way equivalence. The following conditions are equivalent:

Large-q exponential-regime conjecture. For all $q$ sufficiently large, there exists an infinite sequence of $d$-regular graphs with $\lambda_2=\Theta(1/\sqrt{d})$ for some…

A lattice packing in $\mathbb{R}^4$ has density bounded above by the lattice three-point bound, obtained from a semidefinite-programming formulation of the three-point bound. Sharp…

Let $\omega_N\subset\mathbb{S}^2$ be a collection of $N$ antipodal points, meaning that $p\in\omega_N$ implies $-p\in\omega_N$. Let $\mathscr{E}_{\log}(\omega_N)$ denote its logari…

Strengthened packing conjecture. In the conclusion of the hexagonal packing uniqueness conjecture, the assertion $\delta(S)>0$ can be replaced by the stronger assertion…

Let $f$ and $\widehat{f}$ be optimal radial functions for the Euclidean linear programming sphere-packing bound, normalized so that … Their quadratic Taylor coefficients are the co…

E8 and Leech limiting-ratio conjecture. In the $E_8$ case,

Let $f$ and $\widehat f$ be the limiting functions from the convergence conjecture, and call the prescribed zeros of the construction the forced roots. No-unwanted-roots conjecture…

Let $f_k$ and $\widehat f_k$ be the auxiliary functions obtained by forcing the modified roots described in the paper, viewed as analytic functions of a radial variable. Convergenc…

Let $n\in\{2,8,24\}$, and let $f$ satisfy the hypotheses of the Cohn–Elkies linear-programming bound. Write $r$ for the radius parameter in that bound. Cohn–Elkies sphere-packing c…

In three-dimensional space, consider packings of congruent non-overlapping spheres, and let their density be the proportion of space occupied by the spheres, interpreted as the asy…

Let $f \colon [0,\infty) \to \mathbb{R}$ be continuous and integrable on $\mathbb{H}^n$, and suppose … while $\widehat{f}(\lambda) \ge 0$ for all $\lambda>0$ and $\widehat{f}(0)>0$…

Let $C_{s,d}$ be the asymptotic Riesz-energy constant defined, for $d\geq 2$, by … Here $\mathcal{E}_s([0,1]^d;N)$ denotes the optimal discrete Riesz $s$-energy of $N$ points in th…