Tiling By Translates Of A Function: Results And Open Problems

We say that a function f is an element of L-1(R) tiles at level w by a discrete translation set Lambda subset of R, if we have E-lambda is an element of Lambda f(x - lambda) = w a.e. In this paper we survey the main results, and prove several new ones, on the structure of tilings of R by translates of a function. The phenomena discussed include tilings of bounded and of unbounded density, uniform distribution of the translates, periodic and non-periodic tilings, and tilings at level zero. Fourier analysis plays an important role in the proofs. Some open problems are also given.