Number Theoretic Related to the Scaling Spectrum of Self-similar Measure with Three Element Digit Sets

摘要

Let q > 1 be an integer, and let D = (0, a, b) with (a, b) (-1, 1) (mod 3). It is well known that the self-similar measure mu(3q,D) generated by the iterated function system {tau(d)(x) = x+d/3q}(d is an element of D) is a spectral measure with a spectrum @@@ Lambda (3q, C) = {Sigma(n)(j=0) (3q)(j) c(j) : c(j) is an element of C = {-q, 0, q} and n is an element of N}. @@@ In this paper, we study a class of primitive numbers alpha so that we can judge whether alpha Lambda (3q, C) is a spectrum for mu(3q,D). Moreover, by applying the properties of congruences and the order of elements in the finite group, we give some conditions on prime numbers alpha and beta such that the scaling set alpha beta Lambda (3q,C) is also a spectrum of mu(3q,D).

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