A Regular Problem

Here's a problem from Janos Simon. For a set A, let L(A)={x | for some m≥0, x^m is in A} Simon asked, as a homework problem, to show that if A is regular then L(A) is also regular. The standard solutions require an exponential increase in the number of states. Simon asks whether this is needed in general. If A is accepted by an n-state DFA, can one find a poly(n)-state DFA for L(A) or is an exponential state DFA for L(A) required?

As an aside the submission deadline for the Complexity Conference is right around the corner. Get your papers ready.