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The aim of this partly expository paper is to present and discuss two classesof sets of integers (Jamison and Kazhdan sets) whose definition and/orproperties are determined or inspired by operator-theoretical properties.Jamison sets first appeared in the study of the relationship between the growthof the sequence of norms of iterates of a bounded linear operator on aseparable Banach space and the size of its unimodular point spectrum. Kazhdansubsets of $\mathbb{Z}$ are particular cases of Kazhdan sets in generaltopological groups, which are especially important as they appear in thedefinition of Property (T). This paper is also intended as a companion to theauthors' paper [C.Badea, S.Grivaux, Kazhdan sets in groups and equidistributionproperties, \emph{J. Funct. Anal.} \textbf{273} (2017), p. 1931 -- 1969], whichundertakes a study of Kazhdan subsets of some classical groups without Property(T). We present here in detail the case of the group $\mathbb{Z}$, which is oneof the most natural examples of groups without Property (T), and which may beuseful to build an intuition of some of the main results of [C.Badea,S.Grivaux, Kazhdan sets in groups and equidistribution properties, op. cit.].Also, the proofs in the case of the group $\mathbb{Z}$ rely solely on toolsfrom basic operator theory and harmonic analysis. Some crucial links betweenJamison and Kazhdan sets in $\mathbb{Z}$ are exhibited, and many examples aregiven.

Sets of integers determined by operator-theoretical properties: Jamison and Kazhdan sets in the group $\mathbb{Z}$