Vol. 9, No. 5, 2016

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On polynomial configurations in fractal sets

Kevin Henriot, Izabella Łaba and Malabika Pramanik

Vol. 9 (2016), No. 5, 1153–1184
Abstract

We show that subsets of ${ℝ}^{n}$ of large enough Hausdorff and Fourier dimension contain polynomial patterns of the form

$\left(x,\phantom{\rule{0.3em}{0ex}}x+{A}_{1}y,\dots ,\phantom{\rule{0.3em}{0ex}}x+{A}_{k-1}y,\phantom{\rule{0.3em}{0ex}}x+{A}_{k}y+Q\left(y\right){e}_{n}\right),\phantom{\rule{1em}{0ex}}x\in {ℝ}^{n},\phantom{\rule{0.3em}{0ex}}y\in {ℝ}^{m},$

where ${A}_{i}$ are real $n×m$ matrices, $Q$ is a real polynomial in $m$ variables and ${e}_{n}=\left(0,\dots ,0,1\right)$.