In this work, we introduce and study the notion of local randomness for compact metric groups. We prove a mixing inequality as well as a product result for locally random groups under an additional dimension condition on the volume of small balls, and provide several examples of such groups. In particular, this leads to new examples of groups satisfying such a mixing inequality. In the same context, we develop a Littlewood–Paley decomposition and explore its connection to the existence of a spectral gap for random walks. Moreover, under the dimension condition alone, we prove a multi-scale entropy gain result à la Bourgain–Gamburd and Tao.
To Gopal Prasad on the occasion of his 75th birthday
"Locally Random Groups." Michigan Math. J. 72 479 - 527, August 2022. https://doi.org/10.1307/mmj/20217213