Equicovering masses in the Euclidean plane

Abstract

Classic mass partition results are about dividing the plane into regions that are equal with respect to one or more measures (masses). We introduce a new concept in which the notion of partition is replaced by that of a cover. In this case we require (almost) every point in the plane to be covered the same number of times. If all elements of this cover are equal with respect to the given masses, we refer to them as equicoverings. To construct equicoverings, we study a natural generalization of $k$-fan partitions, which we call spiral equicoverings. Like $k$-fans, these consist of wedges centered at a common point, but arranged in a way that allows overlapping. Our main result nearly characterizes all reduced positive rational numbers $p/q$ for which there exists a covering by $q$ convex wedges such that every point is covered exactly $p$ times. The proofs use results about centerpoints and combine tools from classical mass partition results, and elementary number theory.