Additive spectra of the 1 4 Cantor measure Palle E. T. Jorgensen, Keri A. Kornelson, and Karen L. Shuman Abstract. In this paper, we add to the characterization of the Fourier spectra for Bernoulli convolution measures. These measures are supported on Cantor subsets of the line. We prove that performing an odd additive translation to half the canonical spectrum for the 1 4 Cantor measure always yields an alternate spectrum. We call this set an additive spectrum. The proof works by connecting the additive set to a spectrum formed by odd multiplicative scaling. 1. Introduction Traditional Cantor sets are generated by iterations of an operation of down- scaling by fractions which are powers of a fixed positive integer. For each iteration in the process, we leave gaps. For example, the best-known ternary Cantor set is formed by scaling down by 1 3 and leaving a single gap in each step. An associated Cantor measure μ is then obtained by the same sort of iteration of scales, and, at each step, a renormalization. In accordance with classical harmonic analysis, these measures may be seen to be infinite Bernoulli convolutions. Our present analysis is motivated by earlier work, beginning with [JP98]. We consider recursive down-scaling by 1 2n for n ∈ N and leave a single gap at each iteration-step. It was shown in [JP98] that the associated Cantor measures μ 1 2n have the property that L 2 (μ 1 2n ) possesses orthogonal Fourier bases of complex ex- ponentials (i.e. Fourier ONBs). More recently, it was shown in [Dai12] that the scales 1 2n are the only values that generate measures with Fourier bases. Given a fixed Cantor measure μ, a corresponding set of frequencies Γ of ex- ponents in an ONB is said to be a spectrum for μ. For example, in the case of recursive scaling by powers of 1 4 , i.e. n = 2, a possible spectrum Γ for L 2 (μ) has the form Γ as shown below in Equation (2.4). A spectrum for a Cantor measure turns out to be a lacunary (in the sense of Szolem Mandelbrojt) set of integers or 2010 Mathematics Subject Classification. 42B05, 28A80, 28C10, 47A10. Key words and phrases. Cantor set, fractal, measure, Bernoulli convolution, spectrum, operator, isometry, unitary. The second author was supported in part by grant #244718 from The Simons Foundation. arXiv:1310.7242v1 [math.SP] 27 Oct 2013