New Proofs of Plünnecke-type Estimates for Product Sets in Groups

Abstract

We present a new method to bound the cardinality of triple product sets in groups and give three applications. A new and unexpectedly short proof of the Plunnecke-Ruzsa sumset inequalities for Abelian groups. A new proof of a theorem of Tao on triple products, which generalises these inequalities when no assumption on commutativity is made. A further generalisation of the Plunnecke-Ruzsa inequalities in general groups.

15 pages Same results as before. Changes in the introduction and presentation

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• New Upper Bound for Sums of Dilates
• Growth in groups: ideas and perspectives
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• On sets with few distinct distances
• Almost homomorphisms between the Boolean cube and groups of prime order
• Sets without $k$-term progressions can have many shorter progressions
• A question of Bukh on sums of dilates
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• Approximate Subloops in Moufang Loops
• If $A+A$ is small then $AAA$ is superquadratic
• The stability of finite sets in dyadic groups
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