This question is a special case of a problem proposed by John Brzozowski in 1966 during the seventh SWAT (now FOCS) Conference. Let $A$ be a finite alphabet and let $A^*$ be the free monoid on $A$. A subset $L$ of $A^*$ has the finite power property if there exists a natural number $n$ for which $L^* = L^n$. Brzozowski's original question was the following:

Given a regular subset $L$ of $A^*$, decide whether or not $L$ has the finite power property.

After some premiminary remarks by Linna [4], the problem was shown to be decidable independently by K. Hashiguchi [1] and Imre Simon [5]. It was also shown to be undecidable for context-free languages [2]. An elegant purely semigroup theoretic proof was given by Kirsten [3].

Hashiguchi's solution is very short and works directly on the automaton recognizing $L$. Imre Simon reduced the finite power property to the finite closure problem for the tropical semiring. (Recall that the adjective tropical was coined in honor of Imre Simon).

The case you consider is $A = \{a\}$, a much simpler case and deciding the finite power property for regular subsets of $\mathbb{N}$ (= finite unions of arithmetic progressions) is likely to be much easier. Context-free and regular subsets coincide in $\mathbb{N}$, but perhaps the problem is still decidable for larger classes.

[1] K. Hashiguchi, A decision procedure for the order of regular events, Theoret. Comput. Sci. 8 (1979) 69–72.

[2] C. B. Hughes and S. M. Selkow. The finite power property for context- free languages. Theoretical Comput. Sci., 15:111-114, 1981.

[3] D. Kirsten, The finite power problem revisited, Information Processing Letters 84 (2002) 291–294

[4] M. Linna, Finite power property of regular languages, in “Automata, Languages and Programming” (M. Nivat, Ed.), pp. 87-98, North-Holland, Amsterdam, 1973.

[5] I. Simon, Limited subsets of a free monoid, in: Proceedings of the 19th IEEE Annual Symposium on Foundations of Computer Science, North Carolina Press, 1978, pp. 143–150.

This question is a special case of a problem proposed by John Brzozowski in 1966 during the seventh SWAT (now FOCS) Conference. Let $A$ be a finite alphabet and let $A^*$ be the free monoid on $A$. A subset $L$ of $A^*$ has the finite power property if there exists a natural number $n$ for which $L^* = L^n$. Brzozowski's original question was the following:

Given a regular subset $L$ of $A^*$, decide whether or not $L$ has the finite power property.

After some preliminary remarks by Linna [4], the problem was shown to be decidable independently by K. Hashiguchi [1] and Imre Simon [5]. It was also shown to be undecidable for context-free languages [2]. An elegant purely semigroup theoretic proof was given by Kirsten [3].

Hashiguchi's solution is very short and works directly on the automaton recognizing $L$. Imre Simon reduced the finite power property to the finite closure problem for the tropical semiring. (Recall that the adjective tropical was coined in honor of Imre Simon).

The case you consider is $A = \{a\}$, a much simpler case and deciding the finite power property for regular subsets of $\mathbb{N}$ (= finite unions of arithmetic progressions) is likely to be much easier. Context-free and regular subsets coincide in $\mathbb{N}$, but perhaps the problem is still decidable for larger classes.

[1] K. Hashiguchi, A decision procedure for the order of regular events, Theoret. Comput. Sci. 8 (1979) 69–72.

[2] C. B. Hughes and S. M. Selkow. The finite power property for context- free languages. Theoretical Comput. Sci., 15:111-114, 1981.

[3] D. Kirsten, The finite power problem revisited, Information Processing Letters 84 (2002) 291–294

[4] M. Linna, Finite power property of regular languages, in “Automata, Languages and Programming” (M. Nivat, Ed.), pp. 87-98, North-Holland, Amsterdam, 1973.

[5] I. Simon, Limited subsets of a free monoid, in: Proceedings of the 19th IEEE Annual Symposium on Foundations of Computer Science, North Carolina Press, 1978, pp. 143–150.