Covering systems

Covering systems were introduced by Paul Erdős [8] in 1950. A covering system is a collection of congruences of the form x = ai(mod mi) whose union is the integers. These can then be specialised to being incongruent (that is, having distinct moduli), or disjoint, in which each integer satisfies exactly one congruence. -- This thesis studies incongruent restricted disjoint covering systems (IRDCS), collections of congruence classes which cover a finite interval of the integers disjointly, subject to an additional technical condition. There exist IRDCS of length 11 and all lengths greater than or equal to 17. These IRDCS are used to study questions analogous to those of interest in covering systems. We focus on the following questions. (1) Can the smallest modulus of some IRDCS be arbitrarily large? (2) Do there exist IRDCS with all moduli odd? (3) What is the appropriate two-dimensional generalisation? This thesis addresses these questions and makes significant headway towards their resolution.-- Chapter 5 studies IRDCS with large minimum modulus. We present, amongst other examples, one IRDCS with minimum modulus 50. -- In Chapter 6 it is shown that there are IRDCS with only odd moduli. The smallest example is one of length 83. This chapter will present information on all of the known examples of what will be referred to as odd IRDCS. -- Finally, in Chapter 7, we extend the definition of IRDCS to two dimensions, determining conditions on the relevant parameters for the existence of such structures. In this chapter we also study some of the structural properties, analogous to those of one-dimensional IRDCS, for these new constructions.