I am told that the Falconer distance conjecture fails trivially in one dimension, but I really cannot find any reference for that. Precisely, I am asking the following question:


For a compact set $E\subseteq \mathbb R^n$, we define its distance set $\Delta(E)\subseteq [0,\infty)$ to be:
$$
\Delta(E)=\{|x-y|:x,y\in E\}.
$$
Then if $n=1$, we ask the following questions?


1. Can we find a compact set $E\subseteq [0,1]$ such that $\mathrm{dim}\_H(E)>1/2$ but $\mathcal L^1(\Delta(E))=0$?
2. Can we find a compact set $E\subseteq [0,1]$ such that $\mathrm{dim}\_H(E)=1$ but $\mathcal L^1(\Delta(E))=0$?



Failure of Falconer distance problem in one dimension
The answer to the second question (and thus the first) is yes, and the reference is Section 4.12 of P. Mattila, *Geometry of Sets and Measures in Euclidean Spaces*, CUP, 1995. The set in question is a Cantor type set, constructed by taking intersections of an increasing number of spaced intervals of decreasing lengths.


See also Falconer's paper [Dimensions of Intersections and Distance Sets for Polyhedral Norms](https://projecteuclid.org/download/pdf_1/euclid.rae/1129416466) which mentions existence of such sets in the proof of Corollary 3.


