A Connection between Fixed-Point Theorems and Tiling Problems

The classic Banach Contraction Principle states that any contraction on a complete metric space has a unique fixed point. Rather than requiring that a single operator be a contraction, we consider a minimum involving a set of powers of that operator and derive fixed-point results. Ordinary analytical techniques would be extremely unwieldy, and so we develop a method for attacking this problem by considering a related problem on tiling the integers.