Vol. 10, No. 7, 2016

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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 ${\mathbb{F}}_{q}^{n}$ 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