The existence of Fourier basis for some Moran measures.

Let { b n } n = 1 ∞ be a sequence of integers bigger than 1 and let { D n } n = 1 ∞ be a sequence of digit sets in Z where D n = { 0 , r n , 2 r n , ⋯ , ( q n − 1 ) r n } . The family of functions { f n , d ( x ) = b n − 1 ( x + d ) : d ∈ D n } n = 1 ∞ is called a Moran iterated function system (IFS) . In this paper we prove that the associated Moran measure generated by an infinite convolution of atomic measures with equal distribution μ { b n } , { D n } = δ b 1 − 1 D 1 ⁎ δ ( b 1 b 2 ) − 1 D 2 ⁎ ⋯ ⁎ δ ( b 1 b 2 ⋯ b n ) − 1 D n ⁎ ⋯ is a spectral measure if r n q n | b n . [ABSTRACT FROM AUTHOR]