The structure of the exponent set for finite cyclic groups

We survey properties of the set of possible exponents of subsets of $\Z_n$ (equivalently, exponents of primitive circulant digraphs on $n$ vertices). Let $E_n$ denote this exponent set. We point out that $E_n$ contains the positive integers up to $\sqrt{n}$, the `large' exponents $\lfloor \frac{n}{3} \rfloor +1, \lfloor \frac{n}{2} \rfloor, n-1$, and for even $n \ge 4$, the additional value $\frac{n}{2}-1$. It is easy to see that no exponent in $[\frac{n}{2}+1,n-2]$ is possible, and Wang and Meng have shown that no exponent in $[\lfloor \frac{n}{3}\rfloor +2,\frac{n}{2}-2]$ is possible. Extending this result, we show that the interval $[\lfloor \frac{n}{4} \rfloor +3, \lfloor \frac{n}{3} \rfloor -2]$ is another gap in the exponent set $E_n$. In particular, $11 \not\in E_{35}$ and this gap is nonempty for all $n \ge 57$. A conjecture is made about further gaps in $E_n$ for large $n$.