Summary: In [J. Anal. Math. 128, 289–335 (2016; Zbl 1375.28008)], V. Chan et al. investigated geometric configurations inside thin subsets of Euclidean space possessing measures with Fourier decay properties. In this paper we ask which configurations can be found inside thin sets of a given Hausdorff dimension without any additional assumptions on the structure. We prove that if the Hausdorff dimension of \(E \subset \mathbb{R}^d\), \(d \geq 2\), is greater than \(\frac{1}{2}(d+1)\) then, for each \(k \in \mathbb{Z}^+\), there exists a nonempty interval \(I\) such that, given any sequence \(\{t_1, t_2, \dots, t_k: t_j \in I\}\), there exists a sequence of distinct points \({\{x^j\}}_{j=1}^{k+1}\) such that \(x^j \in E\) and \(| x^{i+1}-x^i|=t_j\) for \(1 \leq i \leq k\). In other words, \(E\) contains vertices of a chain of arbitrary length with prescribed gaps.