Bourgain-Tzafriri猜想与不可铺投影的具体构造
The Bourgain-Tzafriri conjecture and concrete constructions of non-pavable projections
摘要：
Kadison-Singer问题（KS）和铺路猜想（PC）等价于Bourgain-Tzafriri猜想（BT）。此外，已知（PC）对于具有恒定对角线$1/2$的$2$铺设投影失败。但这个事实的证明是存在性证明。我们将使用离散傅里叶变换矩阵的变化来构造这些投影和具有恒定对角线$1/r$的投影的具体示例，这些投影在很强的意义上不是$r$可铺的。1989年，Bourgain和Tzafriri证明了具有小元素的零对角矩阵类（对于$n$维希尔伯特空间，在$\le 1/log^{1+ε}n$阶上）是{ \em pavable}。人们一直认为这个结果对BT猜想也成立--尽管没有人正式检验过它，但我们将证明事实并非如此。我们将证明，如果BT-猜想对于系数小的向量（在$\le C/\sqrt{n}$的量级）为真，那么BT-猜想为真，因此KS和PC为真。
英文摘要：
It is known that the Kadison-Singer Problem (KS) and the Paving Conjecture (PC) are equivalent to the Bourgain-Tzafriri Conjecture (BT). Also, it is known that (PC) fails for $2$-paving projections with constant diagonal $1/2$. But the proofs of this fact are existence proofs. We will use variations of the discrete Fourier Transform matrices to construct concrete examples of these projections and projections with constant diagonal $1/r$ which are not $r$-pavable in a very strong sense. In 1989, Bourgain and Tzafriri showed that the class of zero diagonal matrices with small entries (on the order of $\le 1/log^{1+ε}n$, for an $n$-dimensional Hilbert space) are {\em pavable}. It has always been assumed that this result also holds for the BT-Conjecture - although no one formally checked it. We will show that this is not the case. We will show that if the BT-Conjecture is true for vectors with small coefficients (on the order of $\le C/\sqrt{n}$) then the BT-Conjecture is true and hence KS and PC are true.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
