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

TL;DR

This paper characterizes when certain fractal measures generated by iterated function systems with alternating contraction ratios are spectral, revealing conditions on parameters for the existence of orthogonal exponential bases.

Contribution

It provides a complete characterization of spectrality for a class of fractal measures with periodic alternating contraction ratios, extending previous results in the field.

Findings

Spectrality occurs if and only if the inverse of the contraction ratio is an integer divisible by a specific parameter.

When not spectral, the measure's $L^2$ space contains at most $s$ mutually orthogonal exponential functions.

The results generalize recent work by Wu on spectral measures with alternating ratios.

Abstract

For a Borel probability measure $\mu$ on $\mathbb{R}^{n}$, it is called a spectral measure if the Hilbert space $L^{2}(\mu)$ 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 $\rho\in(0,1)$, we define the IFS as follows: $$\{\tau_d(\cdot)=(-1)^{\lfloor\frac{d}{m}\rfloor}\rho(\cdot+d)\}_{d\in D_{2Nm}},$$ where $D_k=\{0,1,\cdots,k-1\}$ and $\lfloor x\rfloor$ denotes the floor function. We prove that the associated self-similar measure $\nu_{\rho,D_{2Nm}}$ is a spectral measure if and only if $\rho^{-1}=p\in\mathbb{N}$ and $2Nm\mid p$. Furthermore, for any positive integers $p,s\geq2$, if $m=1$ and $\gcd(p,s)=1$ we show that $\nu_{p^{-1},D_{s}}$ is not a spectral measure and $L^2(\nu_{p^{-1},D_{s}})$ 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].