Covering Systems and Their Connections to Zero - Sums

A finite system of residue classes is called a covering system if every integer belongs to one of the residue classes. Paul Erdős invented this concept and initiated the study of this fascinating topic. On the basis of known connections between covering systems and unit fractions, the speaker recently found that covering systems are closely related to zerosum problems on abelian groups (another interesting topic initiated by P. Erdős). In this talk we will introduce some recent results connecting covering systems and zero-sums, as well as several related open problems. We will also give a survey of results on covers of a group by cosets, and talk about Gao and Geroldinger’s reduction of some zero-sum problems to the study of coverings of a subset of a group, as well as Dimitrov’s new discovery of the connection between the Davenport constant of Zn and covers of Zn. 1. Properties of covers of Z related to unit fractions For a ∈ Z and n ∈ Z = {1, 2, 3, · · · } we call a(n) = a+ nZ = {a+ nx : x ∈ Z} a residue class with modulus n. For a finite system A = {as(ns)}s=1 = {a1(n1), . . . , ak(nk)} 1 2 ZHI-WEI SUN of residue classes, if ⋃k s=1 as(ns) = Z then we call A a covering system of Z, or a cover of Z in short. This concept was first introduced by Paul Erdős in the early 1930’s. Clearly A forms a cover of Z if and only if it covers 0, 1, · · · , NA − 1, where NA is the least common multiple of the moduli n1, · · · , nk. If A covers every integer exactly once, then we call A an exact cover of Z or a disjoint cover of Z. As any integer can be written uniquely in the form nq + r with q ∈ Z and r ∈ R(n) = {0, 1, · · · , n− 1}, the finite system {r(n)}n−1 r=0 is a disjoint cover of Z. The first nontrivial cover of Z with distinct moduli was the following one discovered by P. Erdős. B = {0(2), 0(3), 1(4), 5(6), 7(12)}. Note that NB = 12 and B covers 0, 1, . . . , 11. Clearly each residue class as(ns) in system A covers exactly NA/ns integers in R(NA) = {0, 1, . . . , NA − 1}. Thus, if A is a cover of Z then |R(NA)| 6 k ∑ s=1 NA ns and hence k ∑ s=1 1 ns > 1; if A is a disjoint cover of Z then ∑k s=1 1/ns = 1. Observe that the sum of reciprocals of the moduli in the cover B equals 1 2 + 1 3 + 1 4 + 1 6 + 1 12 = 1 1 3 . In 1989, by using the Riemann zeta function, M. Z. Zhang [J. Sichuan Univ. (Nat. Sci. Ed.)] showed the following surprising result: If system COVERING SYSTEMS AND THEIR CONNECTIONS TO ZERO-SUMS 3 A is a cover of Z then ∑ s∈I 1/ns ∈ Z for some I ⊆ [1, k] = {1, · · · , k}. The starting point of Zhang is that A forms a cover of Z if any only if