Scaling of spectra of self-similar measures with consecutive digits

Number Theory Problems Related to the Spectrum of Cantor-Type Measures With Consecutive Digits

“…In this paper, we study the scaling of spectra of a class of self-similar measures with consecutive digits. Wu et al [23,24] considered the same problem and we extend their work by discussing all integers τ.…”

“…They gave a complete characterisation of the maximal orthogonal sets of µ 4 . Following Wu and Zhu [24], if Λ is a spectrum of µ, the problem of finding all real numbers τ such that τΛ is also a spectrum of µ is called the scaling spectra problem. The scaling spectra problem can be traced back to the question raised first by Fuglede [12] for open subsets Ω in R n concerning orthogonal Fourier bases for the corresponding Hilbert space L 2 (Ω) with respect to Lebesgue measure, that is, orthogonal bases of complex exponentials.…”

Scaling of spectra of a class of self‐similar measures on R

“…We don't give the proof of the following lemma, since it follows readily from the proof of Lemma 2.1 in or the proof of Lemma 4.5 in . Lemma Let be as in Theorem .…”

“…By applying the properties of congruences and the order of elements in the finite group, we prove the following theorem which gives a precise characterization for Λ( , ) to be a spectrum. However, from [25] or [26], we can just see that if is a prime number and ( ) > +1 − 1, then Λ( , ) is a spectrum of , . The next theorem tells us that the condition ( ) > +1 − 1 can be weakened.…”

“…By Theorem 1.3 in [18], we can obtain the following lemma. We don't give the proof of the following lemma, since it follows readily from the proof of Lemma 2.1 in [7] or the proof of Lemma 4.5 in [25].…”