The generalized Fuglede's conjecture holds for a class of Cantor-Moran measures

Suppose b=\bₙ\₍=₁^ is a sequence of integers bigger than 1 and D=\{ D₍\}₍=₁^ is a sequence of consecutive digit sets. Let ₁, { D} be the Cantor-Moran measure defined by eqnarray* ₁, { D}&=& ₁₁䃑{ D₁}₁₁䃑₁䃒{ D₂} ₁₁䃑₁䃒₁䃓{ D₃}. eqnarray* We prove that L² (₁, { D}) possesses an exponential orthonormal basis if and only if ₁, { D}= L₀, ₍䃑/₁䃑 for some Borel probability measure. This theorem shows that the generalized Fuglede's conjecture is true for such Cantor-Moran measure. An immediate consequence of this result is the equivalence between the existence of an exponential orthonormal basis and the integral tiling of Dₙ= D₍+bₙ D₍-₁+b₂ bₙ D₁ for n1.