Spectra for finite unions of line segments

TL;DR

This paper investigates the spectral properties of arc-length measures on unions of line segments, revealing conditions under which spectra are line-like or more complex, and establishing growth bounds for orthogonal sets in these measures.

Contribution

It characterizes spectral measures supported on unions of line segments, showing line spectra for two segments and complexity for three or more, and extends growth bounds to general arc-length measures.

Findings

Spectral measures on two segments have line spectra.

Unions of three or more segments can have non-line spectra.

Orthogonal sets grow at most linearly in radius for these measures.

Abstract

In this paper we study the spectrality of arc-length measures supported on the union of two line segments in the plane. We show that any such spectral measure must admit a line spectrum. Moreover, when the two segments are non-parallel, such spectral measure admits only line spectra. Thus, in this case every spectrum is one dimensional. In addition we show that this property fails for unions of three or more segments in the plane. We construct some arc-length spectral measures supported on the union of at least three line segments such that none of its spectra is contained in a line. Finally, we work in the general framework of arc-length measures supported on finite unions of curves in $\mathbb{R}^d$. We show that the size of any orthogonal set for such a measure inside a ball of radius $R$ grows at most linearly in $R$. We also give an alternative proof of this bound, and in fact obtain a more general result of growth rate of orthogonal sets for Ahlfors--David regular measures in $\mathbb{R}^d$ (not restricted to the one-dimensional setting).