A note on large Kakeya sets

IF 0.5 4区 数学 Q3 MATHEMATICS
M. de Boeck, G. Van de Voorde
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引用次数: 0

Abstract

Abstract A Kakeya set 𝓚 in an affine plane of order q is the point set covered by a set 𝓛 of q + 1 pairwise non-parallel lines. By Dover and Mellinger [6], Kakeya sets with size at least q2 – 3q + 9 contain a large knot, i.e. a point of 𝓚 lying on many lines of 𝓛. We improve on this result by showing that Kakeya set of size at least ≈ q2 – q q $\begin{array}{} \displaystyle \sqrt{q} \end{array}$ + 32 $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$q contain a large knot, and we obtain a sharp result for planes containing a Baer subplane.
关于大型Kakeya电视机的注意事项
在q阶仿射平面上的Kakeya集合𝓚是由q + 1对非平行线的集合所覆盖的点集。通过Dover和Mellinger[6],大小至少为q2 - 3q + 9的Kakeya集合包含一个大的结,即一个点𝓚位于许多条线上。我们改进了这一结果,证明大小至少≈q2 - q q $\begin{array}{} \displaystyle \sqrt{q} \end{array}$ + 32 $\begin{array}{} \displaystyle \frac{3}{2} \end{array}$ q的Kakeya集合包含一个大的结,并且对于包含Baer子平面的平面我们得到了一个清晰的结果。
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来源期刊
Advances in Geometry
Advances in Geometry 数学-数学
CiteScore
1.00
自引率
0.00%
发文量
31
审稿时长
>12 weeks
期刊介绍: Advances in Geometry is a mathematical journal for the publication of original research articles of excellent quality in the area of geometry. Geometry is a field of long standing-tradition and eminent importance. The study of space and spatial patterns is a major mathematical activity; geometric ideas and geometric language permeate all of mathematics.
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