Improved bounds in Weaver and Feichtinger Conjectures

TL;DR

This paper improves bounds related to the Weaver and Feichtinger conjectures, providing sharper constants and optimal asymptotic bounds, advancing understanding in frame theory and the Kadison--Singer problem.

Contribution

It refines the constant in the $KS_2$ conjecture of Weaver and establishes optimal asymptotic bounds for partitions in the Feichtinger conjecture.

Findings

Sharpened the constant in the $KS_2$ conjecture of Weaver

Proved optimal asymptotic bounds on partition sizes in the Feichtinger conjecture

Extended the implications of the Marcus, Spielman, and Srivastava solution to the Kadison--Singer problem

Abstract

We sharpen the constant in the $KS_2$ conjecture of Weaver \cite{We}, which was validated by Marcus, Spielman, and Srivastava \cite{MSS} in their solution of the Kadison--Singer problem. We then apply this result to prove optimal asymptotic bounds on the size of partitions in the Feichtinger conjecture.