Group ring elements with large spectral density

TL;DR

This paper constructs specific group ring elements with spectral measures that decay slowly near zero, demonstrating that existing bounds on spectral density are nearly optimal and providing insights into the Novikov-Shubin invariant.

Contribution

It introduces a method to construct group ring elements with spectral measures exhibiting near-optimal decay rates near zero, impacting the understanding of spectral invariants.

Findings

Spectral measure mu((0,eps)) > C/|log(eps)|^(1+d) for small eps

The Novikov-Shubin invariant of constructed elements is 0

Existing bounds on spectral density are nearly tight

Abstract

Given an arbitrary d>0 we construct a group G and a group ring element S in Z[G] such that the spectral measure mu of S has the property that mu((0,eps)) > C/|log(eps)|^(1+d) for small eps. In particular the Novikov-Shubin invariant of any such S is 0. The constructed examples show that the best known upper bounds on mu((0,eps)) are not far from being optimal.