  In a recent paper, Chan, \L aba, and Pramanik investigated geometric
configurations inside thin subsets of the Euclidean set 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 $E
\subset {\Bbb R}^d$, $d \ge 2$, is greater than $\frac{d+1}{2}$, then there
exists a non-empty interval $I$ such that given any sequence $\{t_1, t_2,
\dots, t_k; t_j \in I\}$, there exists a sequence ${\{x^j\}}_{j=1}^{k+1}$, such
that $x^j \in E$ and $|x^{i+1}-x^i|=t_j$, $1 \leq i \leq k$. In other words,
$E$ contains vertices of a chain of arbitrary length with prescribed gaps.

Classical Analysis and ODEs;Combinatorics;Metric Geometry