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

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].