13 problems

Let $1<p<\infty$ and $1\leq k\leq n$. For a matrix $A$ and matrix subspace ${\mathcal M}$, let $R^{(p,k)}=A-B^{(p,k)}$ range over residuals of best approximations with respect to t…

Let ${\mathcal M}$ be a matrix subspace, let $A$ be a matrix, and let $Y_p$ be the unique best approximation to $A$ in ${\mathcal M}$ with respect to the $c_p$ norm for…

Let $\mathcal{F}$ be a finite family of structured matrices, and let $\epsilon>0$. A query is an evaluation involving the unknown matrix and a query vector, as in the paper's matri…

Beresnevich–Guan–Marstrand–Ramírez–Velani conjecture.

Square-root-degree conjecture. Using arguments of Newman and Rivlin, $N=O(\sqrt{q})$ should be sufficient to accurately approximate $\mathbf{A}^p$.

Approximate semidefinite projection error conjecture. Based on experimental evidence, there should be a small constant $\delta$, perhaps $\delta=2$, such that

Let $m,n$ be positive integers with $(m,n)>1$, and let $\varphi$ be a curve satisfying the paper's generic condition. The paper considers the associated homogeneous-space dynamics,…

Let $d\geq2$. For a finite-dimensional Hilbert space $\mathcal H$, consider unitaries $U_1,\ldots,U_d,V_1,\ldots,V_d\in\mathcal U(\mathcal H)$ whose cross-commutators are small. Fo…

Let $\Uptheta\in\widetilde{\mathcal B}(\mathbb R)$ or $\widetilde{\mathcal M}(\mathbb R)$, and let $({}^{\ast}\mathbb Z^{s})^{\rm sym}(\Uptheta)$ be the symmetric matrix Diophantin…

Let $H_1,H_2,H_3$ be self-dual Hermitian matrices, and let $[H_i,H_j]=H_iH_j-H_jH_i$ denote their commutators. Let $I$ be the identity matrix, and let…

Let $H_1,H_2,H_3$ be real, symmetric matrices, and let $[H_i,H_j]=H_iH_j-H_jH_i$ denote their commutators. Let $I$ be the identity matrix. Real symmetric triple approximation conje…

Let $A$ and $B$ be self-dual, Hermitian matrices, and let $[A,B]=AB-BA$ denote their commutator. For a matrix $T$, write $\lVert T\rVert$ for its operator norm. Lin's conjecture fo…

Let $A$) and $B$ be real, symmetric matrices, and let $[A,B]=AB-BA$ denote their commutator. For a matrix $T$, write $\lVert T\rVert$ for its operator norm. Lin's conjecture for re…