Semiclassical measure of the spherical harmonics by Bourgain on ${\mathbb{S}}^3$

Bourgain used the Rudin-Shapiro sequences to construct a basis of uniformly bounded holomorphic functions on the unit sphere in ${\mathbb{C}}^2$. They are also spherical harmonics (i.e., Laplacian eigenfunctions) on ${\mathbb{S}}^3\subset {\mathbb{R}}^4$. In this paper, we prove that these functions tend to be equidistributed on ${\mathbb{S}}^3$, based on an estimate of the auto-correlation of the Rudin-Shapiro sequences. Moreover, we identify the semiclassical measure associated to these spherical harmonics by the singular measure supported on the family of Clifford tori in ${\mathbb{S}}^3$. In particular, this demonstrates a new localization pattern in the study of Laplacian eigenfunctions.