Ungrouped machines
Combinatorics
A new connection between continued fractions and the Bourgain–Gamburd machine reveals a girth-free variant of this widely-celebrated theorem.
On a girth-free variant of the Bourgain–Gamburd machine
A variant of the Bourgain–Gamburd machine without using any girth bounds is obtained. Also, we find series of applications of the Bourgain–Gamburd machine to problems of Additive Combinatorics, Number Theory and Probability,