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Summary
Let k, b >= 2 be two positive integers. For D = k{0, 1, ..., b - 1}, it is well known that the self-similar measure mu(k,b) defined by mu(k,b)(center dot) = 1/b Sigma(b-1)(i=0) mu(k,b)(kb(center dot) - ki) is a spectral measure with a spectrum @@@ Lambda(kb, C) - {Sigma(finite)(j=0) (kb)(j)c(j) : c(j) is an element of C = {0, 1, ..., b-1}}. @@@ In this paper, by applying the properties of congruences and the order of elements in the finite group, we obtain some conditions on the integer p such that the set p Lambda(kb, C) is also a spectrum for mu(k,b). Moreover, an example is given to explain our theory.
