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KAKEYA-TYPE SETS IN LOCAL FIELDS WITH FINITE RESIDUE FIELD

Part of: Classical measure theory

Published online by Cambridge University Press:  17 February 2016

Robert Fraser
Robert Fraser*
Affiliation:
Department of Mathematics, University of British Columbia, Vancouver, BC, Canada V6T 1Z2 email rgf@math.ubc.ca
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Abstract

We present a construction of a measure-zero Kakeya-type set in a finite-dimensional space $K^{n}$ over a local field with finite residue field. The construction is an adaptation of the ideas appearing in works by Sawyer [Mathematika34(1) (1987), 69–76] and Wisewell [Mathematika51(1–2) (2004), 155–162]. The existence of measure-zero Kakeya-type sets over discrete valuation rings is also discussed, giving an alternative construction to the one over $\mathbb{F}_{\ell }\unicode[STIX]{x27E6}t\unicode[STIX]{x27E7}$ presented by Dummit and Hablicsek [Mathematika59(2) (2013), 257–266].

MSC classification

Primary: 28A75: Length, area, volume, other geometric measure theory
Type
Research Article
Information
Mathematika , Volume 62 , Issue 2 , 2016 , pp. 614 - 629
Copyright
Copyright © University College London 2016 

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