Let AG(n,q) the n-dimensional affine space over a finite field with q elements. A Kakeya set is a point set of AG(n,q) which contains a line in every direction. Is it true that every Kakeya set must have cardinality at least , where is a constant depending only on the dimension , but not on q? This question (the finite field Kakeya problem) was first formulated in an expository paper by Tom Wolff in the late 1990s, and addressed in several subsequent papers (Mockenhaupt-Tao 2004, Tao 2005, Bourgain-Katz-Tao 2004). In this paper the problem is completely solved in the case n=2; more precisely, Kakeya sets of minimal size in AG(2,q) are characterized. Some application of this result to blocking sets is showed
The Finite Field Kakeya Problem
MAZZOCCA, Francesco
2008
Abstract
Let AG(n,q) the n-dimensional affine space over a finite field with q elements. A Kakeya set is a point set of AG(n,q) which contains a line in every direction. Is it true that every Kakeya set must have cardinality at least , where is a constant depending only on the dimension , but not on q? This question (the finite field Kakeya problem) was first formulated in an expository paper by Tom Wolff in the late 1990s, and addressed in several subsequent papers (Mockenhaupt-Tao 2004, Tao 2005, Bourgain-Katz-Tao 2004). In this paper the problem is completely solved in the case n=2; more precisely, Kakeya sets of minimal size in AG(2,q) are characterized. Some application of this result to blocking sets is showedI documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
