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

Suppose b=ₙ\ₙ₌₁∞ is a sequence of integers bigger than 1 and D=\ Dₙ\ₙ₌₁∞ is a sequence of consecutive digit sets. Let μ_ b, D be the Cantor-Moran measure defined by {eqnarray*} μ_{{ b},{ D}}&=& δ_{1/b_1{ D}₁}δ_{1/b_1b_2{ D}₂} δ_{1/b_1b_2b_3{ D}₃}⋯. {eqnarray*} We prove that L²(μ_ b, D) possesses an exponential orthonormal basis if and only if μ_ b, Dν= L[0,N₁/b₁] for some Borel probability measure ν. 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= Dₙ+bₙ Dₙ₋₁+b₂⋯ bₙ D₁ for n≥1.