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Using functional and harmonic analysis methods, we study Kazhdan sets intopological groups which do not necessarily have Property (T). We provide a newcriterion for a generating subset $Q$ of a group $G$ to be a Kazhdan set; itrelies on the existence of a positive number $\varepsilon$ such that everyunitary representation of $G$ with a $(Q,\varepsilon )$-invariant vector has afinite dimensional subrepresentation. Using this result, we give anequidistribution criterion for a generating subset of $G$ to be a Kazhdan set.In the case where $G=\mathbb{Z}$, this shows that if $(n_{k})_{k\ge 1}$ is asequence of integers such that $(e^{2i\pi \theta n_{k}})_{k\ge 1}$ is uniformlydistributed in the unit circle for all real numbers $\theta $ except at mostcountably many, then $\{n_{k}\,;\,k\ge 1\}$ is a Kazhdan set in $\mathbb{Z}$ assoon as it generates $\mathbb{Z}$. This answers a question of Y. Shalom from[B.~Bekka, P.~de la~Harpe, A.~Valette, Kazhdan's property (T), Cambridge Univ.Press, 2008]. We also obtain characterizations of Kazhdan sets in secondcountable locally compact abelian groups, in the Heisenberg groups and in thegroup $\textrm{Aff}_{+}(\mathbb{R})$. This answers in particular a questionfrom [B.~Bekka, P.~de la~Harpe, A.~Valette, Kazhdan's property (T), op. cit.].

Kazhdan sets in groups and equidistribution properties