Date/Time : 17 Aug 2022 14:00
Venue : S17 #04-04 (SR3)
Speaker : Tran Chieu Minh, National University of Singapore

Abstract: (Joint with Jing Yifan) Suppose G is a compact simple Lie group, μ is the normalized Haar measure on G, and A, A^2 ⊆ G are measurable. I will discuss a measure growth gap result where we show that μ(A^2) ≥ (2 + η)μ(A) when μ(A) < c with η and c independent of the choice of G quantitatively determined. This is in contrast with the situation in compact connected solvable Lie group, where the Kemperman’s lower bound μ(A^2) ≥ min{2μ(A), 1} cannot be improved.

I will also survey the connections of this result to sumproduct phenomena, a conjecture by Breuillard and Green, the nonabelian Brunn—Minkowski conjecture, the effective Gleason—Yamabe conjecture, and the Kemperman inverse problem.

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Date/Time : 10 Aug 2022 14:00
Venue : S17 #04-04 (Seminar Room 3)
Speaker : Yifan Jing, University of Oxford

Abstract:  Given natural numbers $s$ and $k$, we say that a finite set $X$ of integers is an additive $B_{s}[k]$ set if for any integer $n$, the number of solutions to the equation \[ n = x_1 + … + x_s, \] with $x_1, …, x_s$ lying in $X$, is at most $k$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{s}[k]$ set analogously. These sets have been studied thoroughly from various different perspectives in combinatorial and additive number theory. In this talk, we will discuss sum-product phenomena for these sets. We show that there is either an exceptionally large additive $B_{s}[k]$ set, or an exceptionally large multiplicative $B_{s}[k]$ set, with $k \ll s$. This is joint work with Akshat Mudgal.

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