Spectral Properties of Sierpinski Measures on Rn

Authors：Dai, Xin-Rong; Fu, Xiao-Ye; Yan, Zhi-Hui^*

Source：Constructive Approximation, 2023.

DOI：10.1007/s00365-023-09654-0

• Sierpinski-type measure
• Spectral measure
• Infinite convolution
• Spectrum

Let R = rho I-n and D = {0, e(1), . . . , e(n)}, where rho > 1 and e(i) is the i-th coordinate vector in R-n. The spectral properties of the n-dimensional Sierpinski measure mu(R,D) has been studied over two decades. In this paper, a special type of spectrum called a typical spectrum for mu(R,D) is considered. We show that rho is an element of(n + 1)N is necessary and sufficient for mu(R,D) to admit a typical spectrum if n + 1 is prime. And some necessary conditions for mu(R,D) to admit a typical spectrum are provided when n + 1 is not a prime number. Furthermore, under the condition on real Hadamard matrix, we prove that mu(R,D) admits a quasi-typical spectrum if and only if rho is an element of 2N. These results show that the spectral properties of the Sierpinski measure mu(R,D) are really different between n + 1 is prime and non-prime. As a corollary, we prove that rho is an element of 2N are the only integers such that mu(R,D) becomes a spectral measure when n = 3.

• Institution
  中山大学; 浙江工业大学