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Spectra for finite unions of line segments

Mihail N. Kolountzakis, Ruxi Shi, Sha Wu

TL;DR

The paper investigates spectral measures supported on finite unions of line segments, establishing that two-segment unions have spectra that are necessarily line-based, and that when the two segments are non-parallel every spectrum is a line spectrum. It shows that the line-spectrum restriction breaks for unions of three or more segments by constructing explicit spectral measures whose spectra are not contained in any line. It provides a sharp growth bound for orthogonal exponentials on finite unions of curves, valid in general and extended to Ahlfors–David regular measures, highlighting how one-dimensional structure constrains spectral sets. The results connect spectrality to projection/tiling properties and offer a flexible framework for growth estimates of spectra in higher-dimensional curve unions.

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).

Spectra for finite unions of line segments

TL;DR

The paper investigates spectral measures supported on finite unions of line segments, establishing that two-segment unions have spectra that are necessarily line-based, and that when the two segments are non-parallel every spectrum is a line spectrum. It shows that the line-spectrum restriction breaks for unions of three or more segments by constructing explicit spectral measures whose spectra are not contained in any line. It provides a sharp growth bound for orthogonal exponentials on finite unions of curves, valid in general and extended to Ahlfors–David regular measures, highlighting how one-dimensional structure constrains spectral sets. The results connect spectrality to projection/tiling properties and offer a flexible framework for growth estimates of spectra in higher-dimensional curve unions.

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 . We show that the size of any orthogonal set for such a measure inside a ball of radius grows at most linearly in . 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 (not restricted to the one-dimensional setting).
Paper Structure (11 sections, 31 theorems, 80 equations, 9 figures)

This paper contains 11 sections, 31 theorems, 80 equations, 9 figures.

Table of Contents

  1. Introduction
  2. Line spectra for arc-length measures supported on two line segments
  3. No line spectra for some arc-length measures supported on more than two line segments
  4. Size of orthogonal sets of exponentials
  5. Spectrality of two line segments
  6. Parallel line segments
  7. Non-parallel line segments
  8. Spectral collections of segments without line spectra
  9. Growth bound for orthogonal sets in balls
  10. Any finite collection of curves
  11. Ahlfors--David regular measures

Key Result

Theorem 1

Given are real numbers $a_1$, $a_2$ and $h_1\neq h_2$. Let $T_1, T_2>0$ and $T_1+T_2=2$. Then the measure $\nu=\frac{1}{2}L_{[a_1, a_1+T_1]}\times\delta_{h_1}+\frac{1}{2} L_{[a_2, a_2+T_2]}\times\delta_{h_2}$ is always spectral with one of its spectra contained in a straight line.

Figures (9)

  • Figure 2: (III): Affine transformation
  • Figure 3: For every $k>0$ there exists a line $L$ through the origin such that the distance between the projections of $I_1$ and $I_2$ onto the line $L$ is $k$ times the sum of their projected lengths.
  • Figure 4: Projecting to an interval.
  • Figure 5: Three kinds of equidistant parallel lines.
  • Figure 6: $f(x)=\frac{\sin \pi x}{ \pi x}$
  • ...and 4 more figures

Theorems & Definitions (63)

  • Remark 1
  • Theorem 1
  • Theorem 2
  • Remark 2
  • Theorem 3
  • Remark 3
  • Theorem 4
  • Theorem 5
  • Theorem 6: Any finite collection of curves
  • Corollary 7: Any finite collection of line segments
  • ...and 53 more