Analysis & PDE 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

DOI: 10.2140/apde.2016.9.1153

Abstract

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

(x, x + A_{1} y, \dots, x + A_{k - 1} y, x + A_{k} y + Q (y) e_{n}), x \in ℝ^{n}, y \in ℝ^{m},

where $A_{i}$ are real $n \times m$ matrices, $Q$ is a real polynomial in $m$ variables and $e_{n} = (0, \dots, 0, 1)$ .

Keywords

configurations in fractals, additive combinatorics

Mathematical Subject Classification 2010

Primary: 11B30

Secondary: 28A80

Milestones

Received: 25 November 2015

Revised: 29 March 2016

Accepted: 29 April 2016

Published: 29 July 2016

Authors

Kevin Henriot
Department of Mathematics University of British Columbia 1984 Mathematics Road Vancouver BC V6T 1Z2 Canada

Izabella Łaba
Department of Mathematics University of British Columbia 1984 Mathematics Road Vancouver BC V6T 1Z2 Canada

Malabika Pramanik
Department of Mathematics University of British Columbia 1984 Mathematics Road Vancouver BC V6T 1Z2 Canada