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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.
UDC:
511.7
Received: November 10, 2017
DOI:
10.1134/S0371968518040167
Fulltext:
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