Some Results On Spectral Conjecture

In1979, Fuglede raised the famous Spectral Conjecture, and attracted many mathematicians. With a deep research on the conjecture, mathemati-cians has got many significant results from then on. Unfortunately, by the endeavor of Fields Medal winner T. Tao. and some other mathematicians, the Spectral Conjecture has been proved not to hold when the dimension is bigger than or equal to3. However, people’s interests are no longer con-fined to the conjecture, and has evolved to the spectral problem, which has been deeply investigated.This thesis chooses two problems from this field:1. Whether the corresponding self-similar measure μρ,D is spectral when the digit set is D=ΩN (?) MΩN.2.To generalize the concept of spectrum and (T1)(T2) conditions to that of polynomials, and to recover the relation between spec-trum and (T1)(T2) conditions.The main results of the thesis consists of two parts. Part one is about the spectral property of self-similar measure when the digit set is D=ΩN (?) MΩN, and we found that when MNlq, μρ,D is spectral. For oth-er five conditions which has been considered in this thesis,μρ,D is not spectral. In part two, we give the definition of the spectrum of a integeral polynomial, and showed that a spectral polynomial corresponds to a spec-tral set when the coefficients of the polynomial is either0or1. Then, we found examples which does not satisfy (T1) or (T2) condition respectively, and thus showed that the relation between spectral property and (T1)(T2) properties may not be that direct. On the other hand, we have worked out some significant results on the relation between spectral polynomials and the functions whose coefficients are all0or1.