Quasi-isometries, phase transitions, and other problems in additive number theory

This is a survey of recent work in combinatorial and additive number theory suggested by a problem of Richard Schwartz in metric geometry and geometric group theory. The central object is a group with an infinite set of generators, and the induced metric. Examples are the additive group of integers, with generators being addition (or subtraction) of {a^j: j=0,1,2,...}, for a given integer a>1. One can ask when two such metrics are bi-Lipschitz equivalent. Some results and many open problems will be discussed.