(joint work with P. Varju)
In contrast to the two dimensional case, in dimension $d \geq 3$
averaging operators on the $d-1$-sphere using finitely many rotations,
i.e. averaging operators of the form $Af(x)= |S|^{-1} \sum_{\theta \in
S} f(s x)$ where $S$ is a finite subset of $SO(d)$, can have a spectral
gap on $L^2$ of the $d-1$-sphere. A result of Bourgain and Gamburd shows
that this holds, for instance, for any finite set of elements in $SO(3)$ with algebraic entries and spanning a dense subgroup.
We prove a new spectral gap result for averaging operators corresponding
to finite subsets of the isometry group of ${\bf R}^d$, which is a semi-direct
product of $SO (d)$ and ${\bf R}^d$, provided the averaging operator
corresponding to the rotation part of these elements have a spectral
gap. This new spectral gap result has several applications, and in
particular (sharpening a previous result by Varju) allows us to prove a
local-central limit theorem for a random walks on ${\bf R}^d$ using the
elements of the isometry group that holds up to an exponentially
small scale, as well as to the study of self similar measures in $d \geq
3$ dimensions.