Spectral gaps from dense generation

Establish a spectral gap for random walks generated by arbitrary dense generators of a connected compact simple Lie group, extending results currently known only under additional algebraicity or related hypotheses.

Background

The paper’s method deliberately avoids assuming a uniform spectral gap for the rotation walk, relying instead on a finite-resolution mixing estimate and an $L^1$ smoothing argument. The authors note that spectral gaps are known for certain dense subgroups generated by algebraic elements, but no corresponding general theorem is available for arbitrary dense generators. This is a foundational unresolved problem connected to quantitative mixing of the rotation action.