| In this thesis, we consider spectral sets and spectral self-affine measures. The main goal is to investigate the possible sets that admit exponential orthogonal basis (spectral sets) from tilings in the L 2-spaces under the Lebesgue measure and self-affine measure.;We first study spectral set and its connection to tiling. Our main effort in this part is to present an elementary method of describing certain spectra and tilings. This enables us to obtain several characteristic results that are connected with the dual Fuglede spectral-set conjecture. Furthermore, we derive the general relationship between spectra and tilings, including a spectral-set criterion and its application to the universal spectrum criterion of Lagarias and Szabo. We answer a question of Lagarias and Wang by giving examples to show that a set O ⊂ Rn with finite positive Lebesgue measure 0 < muL(O) < infinity and Zn {0} ⊆ Z(O) := {u ∈ Rn : c&d4;W (u) = 0} need not tile Rn by translations, although such O gives a multiple tile of Rn . At the end of this part, we point out that the spectral-set duality conjecture and the weak spectral-set conjecture formulated by Lagarias, Reeds and Wang are equivalent.;Next we investigate the spectral self-affine measure mu M,D associated with iterated function system (IFS) {&phis; d(x) = M-1( x + d)}d∈D and its dual IFS {psis(x) = M*x + s}s∈S, where M ∈ Mn( Zn ) is an expanding integral matrix, D and S are finite subsets of Zn of the same cardinality |D| = |S|. Based on the previous research, we begin on the question of determining conditions under which EΛ(M,S ) := {e2pii ⟨lambda,x⟩ : lambda ∈ Λ( M, S)} is an orthogonal basis for L2(mu M,D). We first present some elementary properties of compatible pair. We then obtain an easy check condition for (muM,D, Λ( M, S)) not to be a spectral pair. Using this condition, we show that in the Eiffel Tower or 3-dimensional Sierpinski gasket, the corresponding (muM,D, Λ(M, S)) is not a spectral pair. This answers a question considered by Jorgensen, Pedersen and Strichartz. Further generalization of the given condition is discussed. We also give several examples in the final section to illustrate the spectral pair conditions considered here.;Finally we investigate the muM,D-orthogonality and compatible pair conditions as well as the relations between them. The research here is based on the structure of vanishing sums of roots of unity, and is closely related to the problem of spec tral self-affine measure. In the previous study of spectral pair (muM,D, Λ( M, S)), we know that |D| = |S| ≤ |det(M)| is assumed or implied in the indispensable condition that (M-1D, S) is a compatible pair. The relation between the cardinality |D|(=| S|) and |det(M)| seems to be subtle. In this part we first provide a necessary condition for EΛ to be orthogonal in L2(mu M,D) and for (M-1 D, S) to be a compatible pair. This condition shows that the cardinality |D| cannot be too small, for example, |D| cannot be less than the smallest prime divisor of |det(M)| in order to guarantee that EΛ( M,S) is orthogonal in L 2(muM,D). Under certain conditions, we show that the orthogonality of EΛ( M,S) in L2(mu M,D) also implies that (M-1 D, S) is a compatible pair. In particular, the mu M,D-orthogonality of finite set ES implies that EΛ(M,S ) is orthogonal in L2(mu M,D). We prove that every self-affine measure mu M,D coming from a standard digit set is a spectral measure. Two applications of the compatible pair are given. Further applications of our results as well as future research on the existent problems of the subject are also discussed. |
