Algebra Seminar

Seda Albayrak, Department of Pure Mathematics, University of Waterloo

"A Strong version of Cobham’s theorem"

Cobham’s theorem is a fundamental result in the theory of automatic sequences. It says that if a sequence is both p- and q-automatic with p and q multiplicatively independent integers that are each at least two, then this sequence is ultimately periodic. Using automatic sequences, one can define automatic sets. A long-known dichotomy for the growth of automatic sets says that they are either sparse (growth function is poly-logarithmically bounded) or non-sparse. We consider sparse automatic sets and show that if S is a sparse p-automatic set and T is a sparse q-automatic set with p and q multiplicatively independent integers that are greater than or equal to two, then the intersection of S and T is finite. In fact, we can find an explicit bound for the cardinality of the intersection in terms of the sizes of the automata that accept the sets S and T.

Zoom meeting: https://us02web.zoom.us/j/87636005367?pwd=T2p6ZkJlSUp1d093K3d6NkhFZEVJdz09