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Sumset and Inverse Sumset Theory for Shannon Entropy

Published online by Cambridge University Press:  22 January 2010

TERENCE TAO
TERENCE TAO*
Affiliation:
Department of Mathematics, UCLA, Los Angeles CA 90095-1555, USA (e-mail: tao@math.ucla.edu)
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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 A − B, 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 A ⊂ G 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) → ∞.

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Type
Paper
Information
Combinatorics, Probability and Computing , Volume 19 , Issue 4 , July 2010 , pp. 603 - 639
Copyright
Copyright © Cambridge University Press 2010

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