“…In the authors previous works together ( [2], [3]), as well as in other places ( [23], [19], [1]), a class of numbers called the far numbers (specifically "far from the dyadic rationals") play a large role in understanding distinct dyadic systems, which are a set of grids with the property that every cube is contained in a cube from one of the grids of roughly the same size. Distinct dyadic systems are highly useful ( [6], [8], [10], [17], [18], [20], [21] to name just a few), and in our recent works we were able to completely characterize them in both R and R n ( [2], [3]). Essentially, far numbers are bounded away from the dyadic numbers on every scale (see (2.7) for a precise definition), and by shifting a dyadic grid by a far number, one can create a distinct dyadic systems for small scale cubes.…”