Equicovering masses in the Euclidean plane

TL;DR

The paper investigates equicoverings in the Euclidean plane, replacing partitions with covers that locally balance multiple masses and introducing convex $(q,p)$-spiral equicoverings to realize a uniform mass fraction $p/q$ across $q$ wedges. It blends centerpoint geometry, a basic $q$-fan spiral construction, and a Buck–Buck-type concurrent-line partition to yield a near-complete classification: nonexistence for $q<2p$ and $q<3p-3$, and existence for $q \ge 3p$ plus several boundary cases, with a remaining gap at $q=3p-3$ for even $p\ge4$ (e.g., $p/q=4/9$). It also demonstrates non-spiral equicoverings, such as an explicit $(8,3)$ configuration, to illustrate limitations of spiral-only approaches and to highlight open problems. Overall, the work extends mass partition theory to covering configurations, connects centerpoint geometry with convex wedge constructions, and points to new directions for non-spiral and boundary-case constructions in geometric combinatorics.

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.

Figures (6)

• Figure 1: A mass that does not admit a spiral equicovering when $q < 3p - 3$.
• Figure 2: (a) The basic construction. (b) The construction for $q \geq 3p$.
• Figure 3: An illustration of Lemma \ref{['lem:BBG']}: for any mass, the plane can be partitioned using three concurrent lines as described.
• Figure 4: (a) The construction for the case $q = 3p - 3$ with $p = 5$. (b) The construction for the case $q = 3p - 1$ with $p = 2$. In (b), the resulting wedges are grouped into two orbits, distinguished by whether they originate at a black or white dot.
• Figure 5: Both figures depict the same construction for case $q=3p-2$, with $p=3$. Figure (a) shows a possibly non-convex $W_7$ in the white orbit. Figure (b) shows that $W_{14}$, in the black orbit, is disjoint from $W_7$, so it must be convex. Each of the remaining $6$-wedges in the black orbit lies in three consecutive regions $R_j$.

Theorems & Definitions (16)

• Theorem 1.1
• Proposition 2.1
• proof
• Proposition 2.2
• proof
• Proposition 2.3
• proof : Proof of Theorem \ref{['main']}
• Lemma 2.1
• Proposition 2.4
• proof