Consider a collection of positive integers $\{a_i\}_{i=1}^m$ and the distribution $p_i=\frac{a_i}{\sum_{i=1}^ma_i}$.

Similarly for the collection $\{a_i\}_{i=1}^{m+1}$ form the distribution $q_i=\frac{a_i}{\sum_{i=1}^{m+1}a_i}$, for the collection $\{a_i\}_{i=1}^{n}$ form the distribution $r_i=\frac{a_i}{\sum_{i=1}^{n}a_i}$ and for the collection $\{a_i\}_{i=1}^{n+1}$ form the distribution $s_i=\frac{a_i}{\sum_{i=1}^{n+1}a_i}$.

Let $m+1\leq n$. Note all collections are subsets of the collection for the distribution $s$.


For the cases $(1)$ $a_{i+1}=a_i+\theta(\log^ka_i)$, $(2)$ $a_{i+1}=a_i+\theta(a_i)$ where $k$ is a postive constant,  is it true that the difference in entropy is strictly decreasing? That is, is $H(q)-H(p)>H(s)-H(r)$ valid? When can one expect difference in Shannon entropy to be dominant in sequences of these types? Clearly in this post https://mathoverflow.net/questions/191391/entropy-dominance, one criteria for negative result is given. Can $a_i$ be any larger or smaller than in the cases given?


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Can we guess $\lim_{n\rightarrow\infty}H(s)=\infty$ in all these cases?

What spacing in $a_i$ would guarantee $\lim_{n\rightarrow\infty}H(s)<c\in\Bbb N$?