Spectrality of a measure consisting of two line segments

TL;DR

This paper investigates when a measure supported on two line segments forms an orthogonal exponential basis, revealing that most cases are non-spectral except for specific rational parameters, and characterizing when spectra lie on a line.

Contribution

It extends previous work by nearly classifying the spectrality of measures on two segments for all parameters, providing new conditions for line spectra.

Findings

Most measures with parameter t in (-0.5, 0) and irrational t are not spectral.

Spectral measures in this setting have spectra contained in a line.

A simple criterion is given for when the spectrum is contained in a line.

Abstract

Take an interval $[t, t+1]$ on the $x$-axis together with the same interval on the $y$-axis and let $\rho$ be the normalized one-dimensional Lebesgue measure on this set of two segments. Continuing the work done by Lai, Liu and Prince (2021) as well as Ai, Lu and Zhou (2023) we examine the spectrality of this measure for all different values of $t$ (being spectral means that there is an orthonormal basis for $L^2(\rho)$ consisting of exponentials $e^{2\pi i (\lambda_1 x + \lambda_2 y)}$). We almost complete the study showing that for $-\frac12<t<0$ and for all $t \notin {\mathbb Q}$ the measure $\rho$ is not spectral. The only remaining undecided case is the case $t=-\frac12$ (plus space). We also observe that in all known cases of spectral instances of this measure the spectrum is contained in a line and we give an easy necessary and sufficient condition for such measures to have a line spectrum.