Author:
MCDONALD ALEX,TAYLOR KRYSTAL
Abstract
AbstractIn this paper we prove that the set
$\{|x^1-x^2|,\dots,|x^k-x^{k+1}|\,{:}\,x^i\in E\}$
has non-empty interior in
$\mathbb{R}^k$
when
$E\subset \mathbb{R}^2$
is a Cartesian product of thick Cantor sets
$K_1,K_2\subset\mathbb{R}$
. We also prove more general results where the distance map
$|x-y|$
is replaced by a function
$\phi(x,y)$
satisfying mild assumptions on its partial derivatives. In the process, we establish a nonlinear version of the classic Newhouse Gap Lemma, and show that if
$K_1,K_2, \phi$
are as above then there exists an open set S so that
$\bigcap_{x \in S} \phi(x,K_1\times K_2)$
has non-empty interior.
Publisher
Cambridge University Press (CUP)
Cited by
1 articles.
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