Furstenberg sets and Furstenberg schemes over finite fields

Ellenberg, Jordan; Erman, Daniel

doi:10.2140/ant.2016.10.1415

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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

DOI: 10.2140/ant.2016.10.1415

Abstract

We give a lower bound for the size of a subset of $F_{q}^{n}$ 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

Vol. 10, No. 7, 2016