Spectra for finite unions of line segments

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).