Deviation from equidistance for one-dimensional sequences

For a finite sequence ( x i ) i = 1 N in the unit interval, we introduce the gap ratio function which measures the size of the maximal gap length relative to all other gap lengths. This function (asymptotically) captures a lot of information about the degree of uniformity of the sequence. We discuss connections to the theories of dispersion, discrepancy, pair correlation statistics and covering numbers. Furthermore, we explicitly calculate the gap ratio function for some important classes of uniformly distributed sequences.