JournalsrmiVol. 35, No. 7pp. 2169–2186

A Plünnecke–Ruzsa inequality in compact abelian groups

  • Pablo Candela

    Universidad Autónoma de Madrid, Spain
  • Diego González-Sánchez

    Universidad Autónoma de Madrid, Spain
  • Anne de Roton

    Université de Lorraine, Vandœuvre-lès-Nancy, France
A Plünnecke–Ruzsa inequality in compact abelian groups cover

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Abstract

The Plünnecke–Ruzsa inequality is a fundamental tool to control the growth of finite subsets of abelian groups under repeated addition and subtraction. Other tools to handle sumsets have gained applicability by being extended to more general subsets of more general groups. This motivates extending the Pl¨unnecke–Ruzsa inequality, in particular to measurable subsets of compact abelian groups by replacing the cardinality with the Haar probability measure. This objective is related to the question of the stability of classes of Haar measurable sets under addition. In this direction the class of analytic sets is a natural one to work with. We prove a Plünnecke–Ruzsa inequality for KK-analytic sets in general compact (Hausdorff) abelian groups. We also discuss further extensions, some of which raise questions of independent interest in descriptive topology.

Cite this article

Pablo Candela, Diego González-Sánchez, Anne de Roton, A Plünnecke–Ruzsa inequality in compact abelian groups. Rev. Mat. Iberoam. 35 (2019), no. 7, pp. 2169–2186

DOI 10.4171/RMI/1116