On a Special Metric in Cyclotomic Fields

Let p be an odd prime and ω be a primitive pth root of unity. In this paper, we introduce a metric on the cyclotomic field K = Q(ω). We prove that this metric has several remarkable properties, such as invariance under the action of the Galois group. Furthermore, we show that points in the ring of integers O_K behave in a highly uniform way under this metric. More specifically, we prove that for a certain hypercube in O_K centered at the origin, almost all pairs of points in the cube are almost equi-distanced from each other when p and N are large enough. When suitably normalized, this distance is exactly 1/^√6.