arXiv:2611.14475 · v1

Explicit restricted isometries with $k=m^{2/3-o(1)}$

具有 k=m^{2/3-o(1)} 的显式受限等距构造

首次提交 2026-11-11 02:37

版本更新 2026-11-11 02:37

雷达观测 2026-11-11 13:07

摘要

We construct an explicit family of real $m\times n$ restricted isometries with $m\to\infty$, $m/n\to 0$, $δ_k\to 0$, and $k=m^{2/3-o(1)}$. We combine Gershgorin's circle theorem with a flat RIP estimate to reduce the problem to choosing a large vertex set in a strongly regular graph with the property that every small subset spans few edges. We select this vertex set using an incidence geometric result due to Zahl. The proof only uses the entrywise absolute value of the Gram matrix despite the conventional wisdom that such proof techniques exhibit a barrier of $m^{1/2}$. We prove that this barrier is actually $m^{2/3}$, and we show that our particular construction exhibits the same barrier.

作者备注：16 pages