Vol. 10, No. 7, 2016

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ISSN: 1944-7833 (e-only)
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Furstenberg sets and Furstenberg schemes over finite fields

Jordan S. Ellenberg and Daniel Erman

Vol. 10 (2016), No. 7, 1415–1436
Abstract

We give a lower bound for the size of a subset of Fqn containing a rich k-plane in every direction, a k-plane Furstenberg set. The chief novelty of our method is that we use arguments on nonreduced subschemes and flat families to derive combinatorial facts about incidences between points and k-planes in space.

Keywords
Kakeya sets, Furstenberg sets
Mathematical Subject Classification 2010
Primary: 14G15
Secondary: 13P10, 42B25, 51E20
Milestones
Received: 20 April 2015
Revised: 3 May 2016
Accepted: 13 June 2016
Published: 27 September 2016
Authors
Jordan S. Ellenberg
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706
United States
Daniel Erman
Department of Mathematics
University of Wisconsin
480 Lincoln Drive
Madison, WI 53706
United States