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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 $\mathcal L^1(\Delta(E))=0$?

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    $\begingroup$ For an explicit example of a set satisfying 2, let $E$ be the set of points such that the $10^k$th digit of the decimal expansion is 0 for each $k$. $\endgroup$
    – Anthony Quas
    Commented Mar 7, 2019 at 16:38
  • $\begingroup$ Actually for arbitrary $s\in [0,1]$, there is some $E$ so that $s=\mathrm{dim}_H(E)=\mathrm{dim}_H(\Delta(E))$ $\endgroup$
    – 喻 良
    Commented Jul 3 at 8:50

1 Answer 1

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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 which mentions existence of such sets in the proof of Corollary 3.

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