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 Tr. Mat. Inst. Steklova, 2018, Volume 303, Pages 209–238 (Mi tm3954)

An inverse theorem for an inequality of Kneser

T. Tao

Department of Mathematics, University of California, Los Angeles, 405 Hilgard Ave, Los Angeles, CA 90095, USA

Abstract: Let $G = (G,+)$ be a compact connected abelian group, and let $\mu _G$ denote its probability Haar measure. A theorem of Kneser (generalising previous results of Macbeath, Raikov, and Shields) establishes the bound $\mu _G(A + B) \geq \min (\mu _G(A)+\mu _G(B),1)$ whenever $A$ and $B$ are compact subsets of $G$, and $A+B := \{a+b: a \in A, b \in B\}$ denotes the sumset of $A$ and $B$. Clearly one has equality when $\mu _G(A)+\mu _G(B) \geq 1$. Another way in which equality can be obtained is when $A = \phi ^{-1}(I)$ and $B = \phi ^{-1}(J)$ for some continuous surjective homomorphism $\phi : G \to \mathbb{R} /\mathbb{Z}$ and compact arcs $I,J \subset \mathbb{R} /\mathbb{Z}$. We establish an inverse theorem that asserts, roughly speaking, that when equality in the above bound is almost attained, then $A$ and $B$ are close to one of the above examples. We also give a more “robust” form of this theorem in which the sumset $A+B$ is replaced by the partial sumset $A +_{\varepsilon} B := \{1_A * 1_B \geq \varepsilon \}$ for some small $\varepsilon >0$. In a subsequent paper with Joni Teräväinen, we will apply this latter inverse theorem to establish that certain patterns in multiplicative functions occur with positive density.

 Funding Agency Grant Number Simons Foundation National Science Foundation DMS-1266164 The author was supported by a Simons Investigator grant, the James and Carol Collins Chair, the Mathematical Analysis & Application Research Fund Endowment, and by NSF grant DMS-1266164.

DOI: https://doi.org/10.1134/S0371968518040167

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English version:
Proceedings of the Steklov Institute of Mathematics, 2018, 303, 193–219

Bibliographic databases:

UDC: 511.7
Received: November 10, 2017

Citation: T. Tao, “An inverse theorem for an inequality of Kneser”, Harmonic analysis, approximation theory, and number theory, Collected papers. Dedicated to Academician Sergei Vladimirovich Konyagin on the occasion of his 60th birthday, Tr. Mat. Inst. Steklova, 303, MAIK Nauka/Interperiodica, Moscow, 2018, 209–238; Proc. Steklov Inst. Math., 303 (2018), 193–219

Citation in format AMSBIB
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