On non-quasianalytic representations of Abelian groups

We study representations $T_g$ of a locally compact Abelian group $G$ with a scattered spectrum satisfying the conditions: there exists $S \subset G$ such that $G=S-S$ and for all $s\in S$
$$ \|T_{ns}\|=o(n^k), \ \ k \ge 1, \ \ \ln\|T_{-ns}\|=o({\sqrt{n}}), \ \ \text{as}\ n\to+\infty. $$