A simple proof of a theorem of Jean Bourgain

An Elementary Proof of the Restricted Invertibility Theorem

We give an elementary proof of a generalization of Bourgain and Tzafriri's Restricted Invertibility Theorem, which says roughly that any matrix with columns of unit length and bounded operator norm has a large coordinate subspace on which it is well-invertible. Our proof gives the tightest known form of this result, is constructive, and provides a deterministic polynomial time algorithm for finding the desired subspace.

The uniform invariant approximation property for compact groups

In this short note we give a proof of the refined version of the uniform invariant approximation property for compact (non-commutative) groups following the Bourgain's approach.

二维Błocki、$L^p$-Mahler和Bourgain猜想

Two-dimensional Błocki, $L^p$-Mahler, and Bourgain conjectures

本文在二维情况下验证了Błocki关于管域Bergman核下界猜想，同时证明了更广泛的$L^p$-Mahler猜想，揭示了Bourgain强超平面猜想在二维中的简单证明。

A low-technology estimate in convex geometry

Let $K$ be an $n$-dimensional symmetric convex body with $n \ge 4$ and let $K\dual$ be its polar body. We present an elementary proof of the fact that $$(\Vol K)(\Vol K\dual)\ge \frac{b_n^2}{(\log_2 n)^n},$$ where $b_n$ is the volume of the Euclidean ball of radius 1. The inequality is asymptotically weaker than the estimate of Bourgain and Milman, which replaces the $\log_2 n$ by a constant. However, there is no known elementary proof of the Bourgain-Milman theorem.

Bourgain's discretization theorem

Bourgain's discretization theorem asserts that there exists a universal constant $C\in (0,\infty)$ with the following property. Let $X,Y$ be Banach spaces with $\dim X=n$. Fix $D\in (1,\infty)$ and set $δ= e^{-n^{Cn}}$. Assume that $\mathcal N$ is a $δ$-net in the unit ball of $X$ and that $\mathcal N$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $D$. Then the entire space $X$ admits a bi-Lipschitz embedding into $Y$ with distortion at most $CD$. This mostly expository article is devoted to a detailed presentation of a proof of Bourgain's theorem. We also obtain an improvem…

The random paving property for uniformly bounded matrices

This note presents a new proof of an important result due to Bourgain and Tzafriri that provides a partial solution to the Kadison--Singer problem. The result shows that every unit-norm matrix whose entries are relatively small in comparison with its dimension can be paved by a partition of constant size. That is, the coordinates can be partitioned into a constant number of blocks so that the restriction of the matrix to each block of coordinates has norm less than one half. The original proof of Bourgain and Tzafriri involves a long, delicate calculation. The new proof relies on the systemati…

切片定理的一个确定性证明

A Deterministic Proof of the Slicing Theorem

该研究给出Bourgain切片定理的确定性证明，未采用随机相关方法，而是基于测度的确定性单参数二次扰动等内容完成，为该定理提供了全新的非随机证明途径。

Uniqueness of unconditional bases in c_0-products

We give counterexamples to a conjecture of Bourgain, Casazza, Lindenstrauss and Tzafriri that if X has a unique unconditional basis (up to permutation) then so does c_0(X). In particular, we show that for Tsirelson's space T, every unconditional basis of c_0(T) must be equivalent to a subsequence of the canonical basis but c_0(T) still fails to have a unique unconditional basis. We also give some positive results including a simpler proof that c_0(l_1)has a unique unconditional basis.

给出Kannan-Lovasz-Simonovits猜想的证明

Presenting a proof of the Kannan-Lovasz-Simonovits conjecture

本文给出了高维几何中KLS猜想的证明，关键利用Song和Zhang关于倾斜平均高阶导数的准则。

On the vanishing of reduced 1-cohomology for Banach representations

A theorem of Delorme states that every unitary representation of a connected Lie group with nontrivial reduced first cohomology has a finite-dimensional subrepresentation. More recently Shalom showed that such a property is inherited by cocompact lattices and stable under coarse equivalence among amenable countable discrete groups. We give a new geometric proof of Delorme's theorem which extends to a larger class of groups, including solvable $p$-adic algebraic groups, and finitely generated solvable groups with finite Prüfer rank. Moreover all our results apply to isometric representations in…