Vol. 11, No. 5, 2018

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Large sets avoiding patterns

Robert Fraser and Malabika Pramanik

Vol. 11 (2018), No. 5, 1083–1111

We construct subsets of Euclidean space of large Hausdorff dimension and full Minkowski dimension that do not contain nontrivial patterns described by the zero sets of functions. The results are of two types. Given a countable collection of v-variate vector-valued functions fq : (n)v m satisfying a mild regularity condition, we obtain a subset of n of Hausdorff dimension m(v 1) that avoids the zeros of fq for every q. We also find a set that simultaneously avoids the zero sets of a family of uncountably many functions sharing the same linearization. In contrast with previous work, our construction allows for nonpolynomial functions, as well as uncountably many patterns. In addition, it highlights the dimensional dependence of the avoiding set on v, the number of input variables.

geometric measure theory, configurations, Hausdorff dimension, Minkowski dimension
Mathematical Subject Classification 2010
Primary: 28A78, 28A80, 26B10, 05B30
Received: 9 September 2016
Revised: 8 June 2017
Accepted: 2 January 2018
Published: 11 April 2018
Robert Fraser
University of British Columbia
Vancouver, BC
Malabika Pramanik
University of British Columbia
Vancouver, BC