Scaling spectrum of a class of self-similar measures with product form on ℝ

Abstract

Let p, q, N ≥ 2 {N\geq 2} be three positive integers and let D = { 0 , 1 , … , N - 1 } ⊕ N p ⁢ { 0 , 1 , … , N - 1 } {D=\{0,1,\ldots,N-1\}\oplus N^{p}\{0,1,\ldots,N-1\}} be a product form digit set. It is well known that if q ∤ p {q\nmid p} , then the self-similar measure μ N q , D {\mu_{N^{q},D}} generated by the iterated function system { ( N q ) - 1 ⁢ ( x + d ) } d ∈ D , x ∈ ℝ {\{(N^{q})^{-1}(x+d)\}_{d\in D,x\in\mathbb{R}}} is a spectral measure with a spectrum Λ ⁢ ( N q , C ) = { ∑ i = 0