A class of spectral measures with $m$-alternate contraction ratios in $\mathbb{R}$

TL;DR

This work analyzes the spectrality of a family of 1D self-similar measures generated by an IFS with $m$-periodic alternating contraction ratios. By transforming the problem to a Moran-type framework and a uniform-contraction model, the authors derive a complete characterization: the measure is spectral if and only if $\rho^{-1}=p\in\mathbb{N}$ and $2Nm\mid p$, with a corresponding (2-stage) product-form Hadamard triple, and for $m=1$ with $\gcd(p,s)=1$ the measure is nonspectral with at most $s$ orthogonal exponentials. They provide a detailed stepwise argument (Steps A–D) and decompose the digit set to connect to Hadamard-triple machinery; the results generalize prior work on alternating contractions. The paper also analyzes nonspectrality in the $m=1$ case and discusses open problems for odd-digit IFS and related symmetric constructions, contributing to a deeper understanding of spectral fractal measures in one dimension.

Abstract

For a Borel probability measure $μ$ on $\mathbb{R}^{n}$, it is called a spectral measure if the Hilbert space $L^{2}(μ)$ admits an orthogonal basis of exponential functions. In this paper, we study the spectrality of fractal measures generated by an iterated function system (IFS) with $m$-periodic alternating contraction ratios. Specifically, for fixed $m,N\in\mathbb{N}^{+}$ and $ρ\in(0,1)$, we define the IFS as follows: $$\{τ_d(\cdot)=(-1)^{\lfloor\frac{d}{m}\rfloor}ρ(\cdot+d)\}_{d\in D_{2Nm}},$$ where $D_k=\{0,1,\cdots,k-1\}$ and $\lfloor x\rfloor$ denotes the floor function. We prove that the associated self-similar measure $ν_{ρ,D_{2Nm}}$ is a spectral measure if and only if $ρ^{-1}=p\in\mathbb{N}$ and $2Nm\mid p$. Furthermore, for any positive integers $p,s\geq2$, if $m=1$ and $\gcd(p,s)=1$ we show that $ν_{p^{-1},D_{s}}$ is not a spectral measure and $L^2(ν_{p^{-1},D_{s}})$ contains at most $s$ mutually orthogonal exponential functions. These results generalize recent work of Wu [25] [H.H. Wu, Spectral self-similar measures with alternate contraction ratios and consecutive digits, Adv. Math., 443 (2024), 109585].

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Lemma 2.1: JP98
• Theorem 2.2: Dai-He-Lau2014
• Remark 2.3
• Lemma 2.4: An-Wang2021
• Theorem 2.5: Wang-Wang-Dong-Zhang2018
• Theorem 2.6: An-Wang2021