Polynomial bounds for large Bernoulli sections of $\ell_1^N$

Weak equivalence to Bernoulli shifts for some algebraic actions

Namely, we prove that if $G$ is a countable, discrete group and $f\in M_{n}(\Z(G))$ is invertible on $\ell^{2}(G)^{\oplus n},$ but $f$ is not invertible in $M_{n}(\Z(G))$, then the measure-preserving action of $G$ on $X_{f}$ equipped with the Haar measure is weakly equivalent to a Bernoulli action. We shall in fact prove this weak equivalence in the case that $f$ has a "formal inverse in $\ell^{2}$".

Slicing $\ell_p$-balls reloaded: stability, planar sections in $\ell_1$

We show that the two-dimensional minimum-volume central section of the $n$-dimensional cross-polytope is attained by the regular $2n$-gon. We establish stability-type results for hyperplane sections of $\ell_p$-balls in all the cases where the extremisers are known. Our methods are mainly probabilistic, exploring connections between negative moments of projections of random vectors uniformly distributed on convex bodies and volume of their sections.

Limit theorems for the volumes of small codimensional random sections of $\ell_p^n$-balls

We establish Central Limit Theorems for the volumes of intersections of $B_{p}^n$ (the unit ball of $\ell_p^n$) with uniform random subspaces of codimension $d$ for fixed $d$ and $n\to \infty$. As a corollary we obtain higher order approximations for expected volumes, refining previous results by Koldobsky and Lifschitz and approximations obtained from the Eldan--Klartag version of CLT for convex bodies. We also obtain a Central Limit Theorem for the Minkowski functional of the intersection body of $B_p^n$, evaluated on a random vector distributed uniformly on the unit sphere.

$\ell_p$-球的欧几里得截面的半径

Radii of Euclidean sections of $\ell_p$-balls

研究$\ell_p$-球的欧几里得截面半径，探讨其在理论计算机科学中的应用及开放问题

随机截面 $\ell_p$-椭球、最优恢复与对角算子的 Gelfand 数

Random sections of $\ell_p$-ellipsoids, optimal recovery and Gelfand numbers of diagonal operators

本文研究 $\ell_p$-椭球随机截面的外接半径，给出其上界，并证明在 $1\le p\le 2$ 且半轴多项式衰减时，随机信息半径的衰减行为与最优信息半径一致或受常数下界约束，推广了 $p=2$ 的结果。