Event № 688

The evens and odds form a partition of the integers into arithmetic progressions. It is natural to try to describe in general how the integers can be partitioned into arithmetic progressions. For example, a classic result from the 1950's shows that if a set of arithmetic progressions partitions the integers, there must be two arithmetic progressions with the same difference. Another direction is to try to determine when such a partition is a proper refinements of another non-trivial partition.

In my talk I will give some of the more interesting results on this subject, report some (relatively) new results and present two generalizations of partitioning the integers by arithmetic progressions, namely:

1. Partitions of the integers by Beatty sequences (will be defined).

2. Coset partition of a group.

The main conjecture in thefirst topic is due to A. Fraenkel and describes all the partitionshaving distinct moduli. The main conjecture in the second topic, dueto M. Herzog and J. Schonheim, claims that in every coset partition of a group there must be two cosets of the same index.

Again, we will briefly discuss the history of these conjectures, recall some of the main results and report some new results.

Based on joint projects with Y. Ginosar, L. Margolis and J. Simpson.