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Sums, Products and Projections of Discretized Sets

Abstract : In the discretized setting, the size of a set is measured by its covering number by δ-balls (a.k.a. metric entropy), where δ is the scale. In this document, we investigate combinatorial properties of discretized sets under addition, multiplication and orthogonal projection. There are three parts. First, we prove sum-product estimates in matrix algebras, generalizing Bourgain's sum-product theorem in the ring of real numbers and improving higher dimensional sum-product estimates previously obtained by Bourgain-Gamburd. Then, we study orthogonal projections of subsets in the Euclidean space, generalizing Bourgain's discretized projection theorem to higher rank situations. Finally, in a joint work with Nicolas de Saxcé, we prove a product theorem for perfect Lie groups, generalizing previous results of Bourgain-Gamburd and Saxcé.
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Submitted on : Wednesday, January 10, 2018 - 1:25:08 PM
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Weikun He. Sums, Products and Projections of Discretized Sets. Combinatorics [math.CO]. Université Paris-Saclay, 2017. English. ⟨NNT : 2017SACLS335⟩. ⟨tel-01680114⟩

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