When does a Bernoulli convolution admit a spectrum?

Spectrality of a class of infinite convolutions

“…Among those, one of the best known results is that if ρ = 1/q for some integer q > 1, then μ ρ,{0,1} is a spectral measure if and only if q is an even integer [16]. In [2], Dai characterized that these μ 1/q,{0,1} are the only spectral measures among the Bernoulli convolutions.…”

“…In another direction, Jorgensen and Pedersen [16] made a head start to study the spectral property of the self-similar measures. Nowadays, there is a large literature on this topic [1][2][3][4][5][6][7]9,[12][13][14][15][16][19][20][21][23][24][25][26][28][29][30]. Among those, one of the best known results is that if ρ = 1/q for some integer q > 1, then μ ρ,{0,1} is a spectral measure if and only if q is an even integer [16].…”

Spectrality of the planar Sierpinski family

“…For example, in 1998 Jorgensen and Pedersen [16] found the first singular, non-atomic, self-similar spectral measure supported on 1 4 -Cantor set. Nowadays, there is a large literature on this topic [1][2][3][4][5][6][7][8]10,[13][14][15][16]19,20,26,28,29,31]. Most of this literature deals with the issue in one dimension.…”

Spectral property of a class of Moran measures on R

“…In recent years, there has been a wide range of interests in expanding the classical Fourier analysis to fractal or more general probability measures after the pioneer work of Jorgensen and Pedersen [16] in 1998 [1][2][3][4][5][6][7][9][10][11]14,15,[19][20][21][22][23][24][27][28][29]31]. One of the central themes of this area of research involves constructing Fourier bases in L 2 (μ), where μ is a measure which is generated by the iterated function systems.…”