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Nonspectrality of Certain Self-Affine Measures on

DOI: 10.1155/2014/294182

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We will determine the nonspectrality of self-affine measure corresponding to , , and in the space is supported on , where , and are the standard basis of unit column vectors in , and there exist at most mutually orthogonal exponential functions in , where the number is the best. This generalizes the known results on the spectrality of self-affine measures. 1. Introduction Let be an expanding integer matrix; that is, all the eigenvalues of the integer matrix have modulus greater than 1. Associated with a finite subset , there exists a unique nonempty compact set such that . More precisely, is an attractor (or invariant set) of the affine iterated function system (IFS) . Denote by the cardinality of . Relating to the IFS , there also exists a unique probability measure satisfying For a given pair , the spectrality or nonspectrality of is directly connected with the Fourier transform . From (1), we get where The self-affine measure has received much attention in recent years. The previous research on such measure and its Fourier transform revealed some surprising connections with a number of areas in mathematics, such as harmonic analysis, number theory, dynamical systems, and others; see [1, 2] and references cited therein. The and are all determined by the pair . So, for , in the way of examples, there are Cantor set and Cantor measure on the line. And for there is a rich variety of geometries, see Li [3–6], of which the best known example is the Sierpinski gasket. But for , it is more complex. The problem considered below started with a discovery in an earlier paper of Jorgensen and Pedersen [7] where it was proved that certain IFS fractals have Fourier bases, and furthermore that the question of counting orthogonal Fourier frequencies (or orthogonal exponentials in ) for a fixed fractal involves an intrinsic arithmetic of the finite set of functions making up the IFS under consideration. For example, if is the middle-third Cantor example on the line, there cannot be more than two orthogonal Fourier frequencies [7, Theorem 6.1], while a similar Cantor example, using instead a subdivision scale 4 (i.e., ), turns out to have an ONB in consisting of Fourier frequencies [7, Theorem 3.4]. With the effort of Jorgensen and Pedersen [7, Example 7.1], Strichartz [8], Li [3], and Yuan [9], the related conclusions discussed that the diagonal elements of are all even or odd, If one of the diagonal elements is even, what about the result? The general case on the spectrality or nonspectrality of the self-affine measure is not known. The present paper is motivated by

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