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A handy inequality in additive combinatorics is the Plünnecke-Ruzsa inequality:
The proof uses graph-theoretic techniques. Setting , we obtain a useful corollary: if has small doubling in the sense that , then we have for all , where is the sum of copies of .
In a recent paper, I adapted a number of sum set estimates to the entropy setting, in which finite sets such as in are replaced with discrete random variables taking values in , and (the logarithm of) cardinality of a set is replaced by Shannon entropy of a random variable . (Throughout this note I assume all entropies to be finite.) However, at the time, I was unable to find an entropy analogue of the Plünnecke-Ruzsa inequality, because I did not know how to adapt the graph theory argument to the entropy setting.
In fact Theorem 2 is a bit “better” than Theorem 1 in the sense that Theorem 1 needed to refine the original set to a subset , but no such refinement is needed in Theorem 2. One corollary of Theorem 2 is that if , then for all , where are independent copies of ; this improves slightly over the analogous combinatorial inequality. Indeed, the function is concave (this can be seen by using the version of Theorem 2 (or (2) below) to show that the quantity is decreasing in ).
Theorem 2 is actually a quick consequence of the submodularity inequality
in information theory, which is valid whenever are discrete random variables such that and each determine (i.e. is a function of , and also a function of ), and and jointly determine (i.e is a function of and ). To apply this, let be independent discrete random variables taking values in . Observe that the pairs and each determine , and jointly determine . Applying (1) we conclude that
and Theorem 2 follows by telescoping series.
The proof of Theorem 2 seems to be genuinely different from the graph-theoretic proof of Theorem 1. It would be interesting to see if the above argument can be somehow adapted to give a stronger version of Theorem 1. Note also that both Theorem 1 and Theorem 2 have extensions to more general combinations of than ; see this paper and this paper respectively.