If $E$ is a compact subset of ${\Bbb R}^d$, of Hausdorff dimension greater than $\frac{d+1}{2}$, then there exists a non-empty open interval $I$ such that, for any $t \in I$, there exist $x^1, x^2, \dots, x^{k+1} \in E$ such that $|x^{j+1}-x^j|=t$, $1 \leq j \leq k$.

A related open question is the following: find a threshold for the Hausdorff dimension of a compact set $E$ such that given any $r>0$, there exist $x,y,z \in E$ such that $|x-z|=r|x-y|$. This can be regarded as a pinned version of the case $k=1$ of Thm. 1.5 in the referenced article, in the sense that the endpoint $x$ is common to both segments whose length is being compared. Similar questions can be raised when $k>1$.

Article
Existence of similar point configurations in thin subsets of $\mathbb{R}^d$

If you plan to formulate more than one problem all sharing the same background (e.g. they are all from the same paper) then please choose "Group", otherwise select "Single" option.

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