The paper is the third part of a series initiated in two papers to appear in [Math. Proc. Cambr. Philos. Soc. 140(3) (2006)]. Let \(G\) be a locally compact abelian group and \(\Gamma\) its dual. The work is devoted to a notion introduced in the first part: given \(U\subset G\), a subset \(E\) of \(\Gamma\) is an \(I_0(U)\) set if there is a discrete measure \(\mu\) supported by \(U\), whose Fourier transforms interpolate a given bounded and continuous \(\varphi\) from \(E\) to the unit circle, with \(\| \mu\| \leq K(E,U)\| \varphi\| \). Moreover, if the infimum of \(K(E\setminus F,U)\) over finite sets \(F\) is bounded over all open \(U\), then \(E\) is said to have bounded \(I_0(U)\) constants. By nature, this notion is clearly closely linked to the Bohr topology. Corollary 3.6 states that when \(G\) is compact, locally connected and connected, then \(\Gamma\) contains an infinite set that is \(I_0(U)\) for all open \(U\), with bounded constants. We recall that a subset \(E\) of a locally compact abelian group \(\Gamma\) is ”\(\varepsilon\)-Kronecker” if every continuous function from \(E\) to the unit circle can be uniformly approximated on \(E\) by a character with error less than \(\varepsilon\). Miscellaneous results and examples are given in this paper. We present here some of them: Corollary 3.4: Let \(\varepsilon\in(0,\sqrt2)\), \(G\) be connected and \(E\) be an \(\varepsilon\)-Kronecker set. Then \((E\cup E^{-1}).\Delta\) is an \(I_0(G)\) for all finite sets \(\Delta\subset\Gamma\) (see their first paper for a closely related result). Theorem 3.10: Let \(E\) be a Hadamard sequence. Then \(E\cup E^{-1}\) is an \(I_0(U)\) for every open \(U\) of the real line. More general examples of \(I_0(U)\) sets are given in the fourth part of the paper. For some other older results on subsets of \(G\) associated (in some sense) to thin subsets of \(\Gamma\), see, for instance, the works of M. Déchamps-Gondim [Ann. Inst. Fourier 22, No.3, 51-79 (1972; Zbl 0273.43010)] (or her habilitation thesis), or of J.-F. Méla [Bull. Soc. Math. Fr., Suppl., Mém. 19, 26-54 (1969; Zbl 0198.09404)]. More recently there appeared several papers of the second named author of the paper under review: see [Colloq. Math. 67, No.1, 147-154 (1994; Zbl 0835.43004)] for instance.