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

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 μ 3 q , D generated by the iterated function system τ d (x) = x + d 3 q d ∈ D is a spectral measure with a spectrum Λ (3 q , C) = ∑ j = 0 n (3 q) j c j : c j ∈ C = { - q , 0 , q } and n ∈ N. In this paper, we study a class of primitive numbers α so that we can judge whether α Λ (3 q , C) is a spectrum for μ 3 q , D . Moreover, by applying the properties of congruences and the order of elements in the finite group, we give some conditions on prime numbers α and β such that the scaling set α β Λ (3 q , C) is also a spectrum of μ 3 q , D. [ABSTRACT FROM AUTHOR]