Runtime complexity of unary languages
Let $L$ be a language in $\mathrm{TIME}(2^{2^n})$ but not in $\mathrm{TIME}(o(2^{2^n}/2^n))$ (such a language exists by the time hierarchy theorem). Define $U = \{1^n : n \in L\}$. Using an algorithm for $L$, we get that $U \in \mathrm{TIME}(2^n + O(n^2))$. Running the same argument backwards, we see that $U \notin \mathrm{TIME}(o(2^n/n))$ (here we use that $O(4^n) \ll 2^{2^n}/2^n$).