Vol. 11, No. 5, 2018

Download this article
Download this article For screen
For printing
Recent Issues

Volume 11
Issue 5, 1083–1342
Issue 4, 813–1081
Issue 3, 555–812
Issue 2, 263–553
Issue 1, 1–261

Volume 10, 8 issues

Volume 9, 8 issues

Volume 8, 8 issues

Volume 7, 8 issues

Volume 6, 8 issues

Volume 5, 5 issues

Volume 4, 5 issues

Volume 3, 4 issues

Volume 2, 3 issues

Volume 1, 3 issues

The Journal
About the Journal
Subscriptions
Editorial Board
Editors’ Interests
Scientific Advantages
Submission Guidelines
Submission Form
Editorial Login
Contacts
Author Index
To Appear
 
ISSN: 1948-206X (e-only)
ISSN: 2157-5045 (print)
Large sets avoiding patterns

Robert Fraser and Malabika Pramanik

Vol. 11 (2018), No. 5, 1083–1111
Abstract

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.

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