Explicit spectral gap for the quaquaversal operator

Abstract: The spectral gap of an operator is the gap between the largest eigenvalue and the rest of the spectrum. In the mid 90s, John Conway and Charles Radin introduced a three dimensional substitution tiling, the Quaquaversal Tiling, with the property that the orientations of its tiles equidistribute faster than what is possible for two dimensional substitution tilings. Conway and Radin showed that the orientations of the tiles were dense in $SO(3)$ and implicity introduced an operator (later explicitly studied by Draco, Sadun, and Van Wieren) whose spectral gap controls the equidistribution rate. Draco, Sadun, and Van Wieren studied the eigenvalues of this operator numerically and conjectured that it has a spectral gap bounded below by approximately $0.0061697$. We exploit a fact, due to Serre, that the group of orientations for this tiling is $2$-arithmetic and follow a strategy similar to Lubotzky, Phillips, and Sarnak's in order to obtain a lower bound of about $0.0061711$, resolving the conjecture.

Mathematics

Audience: researchers in the topic

Combinatorial and additive number theory (CANT 2025)