Sumset and Inverse Sumset Theory for Shannon Entropy

Author:

TAO TERENCE

Abstract

Let G = (G, +) be an additive group. The sumset theory of Plünnecke and Ruzsa gives several relations between the size of sumsets A + B of finite sets A, B, and related objects such as iterated sumsets kA and difference sets AB, while the inverse sumset theory of Freiman, Ruzsa, and others characterizes those finite sets A for which A + A is small. In this paper we establish analogous results in which the finite set AG is replaced by a discrete random variable X taking values in G, and the cardinality |A| is replaced by the Shannon entropy H(X). In particular, we classify those random variables X which have small doubling in the sense that H(X1 + X2) = H(X) + O(1) when X1, X2 are independent copies of X, by showing that they factorize as X = U + Z, where U is uniformly distributed on a coset progression of bounded rank, and H(Z) = O(1).When G is torsion-free, we also establish the sharp lower bound $\Ent(X+X) \geq \Ent(X) + \frac{1}{2} \log 2 - o(1)$, where o(1) goes to zero as H(X) → ∞.

Publisher

Cambridge University Press (CUP)

Subject

Applied Mathematics,Computational Theory and Mathematics,Statistics and Probability,Theoretical Computer Science

Reference13 articles.

1. [11] Tao T. Freiman's theorem for solvable groups. Preprint.

2. Additive Combinatorics

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4. [6] Gyarmati K. , Matolcsi M. and Ruzsa I. A superadditivity and submultiplicativity property for cardinalities of sumsets. Combinatorica, to appear.

5. Product set estimates for non-commutative groups

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