Spectrality of a measure consisting of two line segments

TL;DR

Problem: determine spectrality for the symmetric additive measure ρ formed from two unit-length segments with parameter t. Approach: analyze the zero set of the Fourier transform ŷρ, derive a projection-based line-spectrum criterion, and apply tiling/finite-complexity arguments to constrain potential spectra. Results: achieve a near-complete classification (except t = -1/2); show non-spectrality for irrational t and for -1/2 < t < 0 under arithmetic conditions, prove that any spectrum must satisfy λ₂−λ₁ = k/(2t+1) and that line spectra arise exactly when the projection is spectral; establish a projection-based method to generate line spectra. Significance: clarifies the relationship between spectrality, projection techniques, and tiling for symmetric additive measures and narrows the open case to t = -1/2, advancing understanding of spectral vs. tiling phenomena in low dimensions.

Abstract

Take an interval $[t, t+1]$ on the $x$-axis together with the same interval on the $y$-axis and let $ρ$ 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 $L^2(ρ)$ consisting of exponentials $e^{2πi (λ_1 x + λ_2 y)}$). We almost complete the study showing that for $-\frac12<t<0$ and for all $t \notin {\mathbb Q}$ the measure $ρ$ 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.

Figures (6)

• Figure 1: The symmetric additive measure we consider in this paper. $\rho$ is a probability measure. On each of the two unit-length segments it equals 1/2 Lebesgue measure. The segments may or may not intersect. By symmetry it is enough to consider the cases $t \ge -\frac{1}{2}$.
• Figure 2: The measure $\rho$ in the case $-1/2<t<0$, with the smooth function $f$ used in the proof.
• Figure 3: We project the two line segments onto $L$. If the resulting two intervals tile the line, then they are spectral and so is the measure $\rho$.
• Figure 4: We choose a line $L$ onto which to project $\rho$ so that the projection meeasure is constant on its support.
• Figure 5: These measures all have a spectrum contained in the $x$-axis.

Theorems & Definitions (22)

• Definition 1.1
• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 2.1
• Remark 2.1
• Corollary 4.1
• proof
• Remark 4.1
• Lemma 5.1