Many inapproximability factors are $2^{n^\epsilon}$. I don't know why this is 2 instead of 3, 4, ..., etc.
To be precise, given a problem,
theorem 1 says: It's NP-hard to approximate the problem to $2^{n^\epsilon}$ for any $0 \leq \epsilon < 1$;
theorem 2 says: It's NP-hard to approximate the problem to $3^{n^\delta}$ for any $0 \leq \delta < 1$.
Can we say theorem 2 is tighter?
I know that given $\delta < 1$, we can find $\epsilon < 1$ such that $3^{n^\delta} \in \mathcal O(2^{n^\epsilon})$. However, the definition of approximation factor is not in asymptotics. We should get $3^{n^\delta} \leq 2^{n^\epsilon}$ if we want to say the two theorems are actually same.