On spectral Cantor-Moran measures and a variant of Bourgain's sum of sine problem

Let 
$$\mu _{p,q}$$

 be a self-similar spectral measure with consecutive digits generated by an iterated function system 
$$\{f_i(x)=\frac{x}{p}+\frac{i}{q}\}_{i=0}^{q-1}$$

, where 
$$2\le q\in {{\mathbb {Z}}}$$

 and q|p. It is known that for each 
$$w=w_1w_2\cdots \in \{-1,1\}^\infty :=\{i_1i_2\cdots :~\text {all}~i_k\in \{-1,1\}\}$$

, the set 
$$\begin{aligned} \Lambda _w=\bigg \{\sum _{j=1}^{n}a_j w_j p^{j-1}:a_j\in \{0,1,\ldots ,q-1\},n\ge 1\bigg \} \end{aligned}$$

is a spectrum of 
$$\mu _{p,q}$$

. In this paper, we study the possible real number t such that the set 
$$t\Lambda _w$$

 are also spectra of 
$$\mu _{p,q}$$

 for all 
$$w\in \{-1,1\}^\infty $$

.

On Spectral Eigenvalue Problem of a Class of Self-similar Spectral Measures with Consecutive Digits

Let k,b ≥ 2 be two positive integers. For D = k{0, 1,…,b − 1}, it is well known that the self-similar measure μk,b defined by μk,b(⋅) = 1 b∑i=0b−1μ k,b(kb(⋅) − ki) is a spectral measure with a spec...

Spectral Eigenvalue Problems of Self-Similar Measures with Consecutive Digits

Spectral Eigenmatrix of the Planar Spectral Measures with Four Elements

Let $$q\ge3$$
 be a positive integer, and let $$D=\{0,a,b\}$$
 with $$\gcd(a,b)=1$$
 . For the self-similar measure $$\mu_{q,D}$$
 which is generated by the iterated function system $$\{\tau_d(x)=\frac{x+d}{q}\}_{d\in D}$$
 , it is well known that if $$q$$
 is divisible by 3 and $$\{a,b\}\equiv\{1,2\}\pmod 3$$
 , $$\mu_{q,D}$$
 is a spectral measure with a spectrum
 $$\Lambda(q,C)=\Big\{\sum_{j=0}^{n}q^ja_j:a_j\in C=\{0,q/3,2q/3\} {\rm and} n \in \mathbb{N} \Big\}.$$
 In this paper, we study some positive integers $$p$$
 such that $$p\Lambda(q,C)$$
 is also a spectrum of $$\mu_{q,D}$$
 .

Multiple spectra of self-similar measures with three digits on $$\mathbb{R}$$ R

We investigate some relations between number theory and spectral measures related to the harmonic analysis of a Cantor set. Specifically, we explore ways to determine when an odd natural number m generates a complete or incomplete Fourier basis for a Cantor-type measure with scale g.

Number Theoretic Considerations Related to the Scaling of Spectra of Cantor-Type Measures

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Number Theoretic Considerations Related to the Scaling of Spectra of Cantor-Type Measures