Spectrality of certain Moran measures with three-element digit sets

Let D-n = {0, a(n), b(n)} = {0,1, 2}(mod 3), p(n) is an element of 3Z(+), n >= 1, satisfy sup(n >= 1) max{vertical bar a(n)vertical bar, vertical bar b(n)vertical bar}/p(n) < infinity. It is well-known that there exists a unique Borel probability measure mu{p(n) }, {D-n} generated by the following infinite convolution product mu{p(n) }, {D-n} = delta(p1-1D1)* delta((p1p2)-1 D2) *... in the weak convergence. In this paper, we give some conditions to ensure that there exists a discrete set Lambda such that the exponential function system {e(2 pi i lambda x)}lambda is an element of Lambda forms an orthonormal basis for L-2(mu{p(n) }, {D-n}). (C) 2017 Elsevier Inc. All rights reserved.