The spectral eigenvalues of a class of product-form self-similar spectral measure
Yan Xiao-yu, Ai Wen-hui
TL;DR
The paper analyzes spectral measures $\mu_{M,D}$ generated by $M=RN^q$ and a product-form digit set $D$, showing $\mu_{M,D}$ is spectral with a model spectrum $\Lambda$ built from a finite-deterministic digit set $L$. It then fully resolves two spectral eigenvalue problems: (i) when a scaled spectrum $t\Lambda$ remains a spectrum (characterized by $\gcd(t,N)=1$ and a divisibility condition on $t$ and words from $L$), and (ii) when there exists a countable $\Lambda'$ such that both $\Lambda'$ and $t\Lambda'$ are spectra for certain rational $t$ with $\gcd(t_i,N)=1$, using infinite-word constructions and Hadamard-triple towers. The methods combine Fourier-analytic structure of $\mu_{M,D}$, Hadamard-triple compatibility, and infinite-convolution techniques to generate new spectra and prove necessary and sufficient criteria. The results extend prior work on spectral measures with product-form digits and contribute to the understanding of spectral tiling and Fourier series convergence on singular self-similar measures.
Abstract
Let μ_{M,D} be the self-similar measure generated by the positive integer M=RN^q and the product-form digit set D=\{0,1,\dots,N-1\}\oplus N^{p_1}\{0,1,\dots,N-1\}\oplus \cdots \oplus N^{p_s}\{0,1,\dots,N-1\}, where R>1, N>1, q, p_i(1\leq i\leq s) are positive integers with gcd(R,N)=1 and p_1<p_2<\cdots<p_s<q. In this paper, we first show that μ_{M,D} is a spectral measure with a model spectrum Λ. Then we completely settle two types of spectral eigenvalue problems for μ_{M,D}. On the first case, for a real t, we give a necessary and sufficient condition under which tΛis also a spectrum of μ_{M,D}. On the second case, we characterize all possible real numbers t such that there exists a countable set Λ'\subset \mathbb{R} such that Λ' and tΛ' are both spectra of μ_{M,D}.