Consider a collection of increasing positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$. Given $\{a_i\}_{i=1}^m$, let $\mathcal{P}_a$ be limiting distribution $m\rightarrow\infty$.

For a given $c>0$, what spacing in $a_i$ would guarantee $H(\mathcal{P}_a)<c$?

What is precise growth of $H(\mathcal{P}_a)$ as a function of gaps of sequences?

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