A Special Case of a Unary Regular Language Containment

Given an n-state unary finite automaton accepting a language T, and an ultimately periodic set S given as a union of arithmetic progressions that can be represented using n^O(1) bits, and whose periods are mutually coprime, deciding whether T ⫅ S is in n^{O(log n)} time. Dropping the mutual coprimality condition, this containment problem becomes NP-hard.