In a recent paper, Chan, Łaba, and Pramanik 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
,
, is greater than
then, for each
, there exists a nonempty
interval
such that, given
any sequence
, there exists a
sequence of distinct points
such that
and
for
. In other
words,
contains vertices of a chain of arbitrary length with prescribed gaps.
Keywords
classical analysis and ODEs, combinatorics, metric
geometry, chains, geometric measure theory, geometric
configurations, Hausdorff dimension, Falconer distance
problem