Nonsingular regret bound for UCB
You can improve this bound by grouping the arms into "almost optimal" arms (arms whose expected payoffs are within a fixed $\Delta$) and the rest of the arms.  So you can bound the regret contributed by the almost optimal arms by $O(T \Delta)$, which actually implies that the worst-case setting of $\Delta$ is approximately $\sqrt{N/T \log T}$, and not a value arbitrarily close to $0$.

This is explained, for example, in section 1.3 of these lecture notes.