Fractal spectral measure theory is a field connecting fractal geometry and Fourier analysis. More specifically, it is about the existence and structure of the exponential orthonormal bases on fractal measures. In this talk, I will introduce some facts about it. Specifically, I will give the proofs of the next three facts about the standard fourth-middle Cantor measure :   is a spectral measure with a spectrum (Jorgensen and Pederson, 1998);  is also a spectrum of (Laba and Wang first noticed this fact); there exists a spectrum of such that its Beurling dimension is zero (Dai, He and Lai, 2013).

If time permits, I will introduce the ideas of proofs about the convergence results of  obtained by Strichartz (2006) and Dutkay et al. (2014).