Variants of the Kakeya problem over an algebraically closed field
The original Kakeya problem asks for the smallest subset of \(\mathbb{R}^2\) that contains a unit line segment in any direction. This problem has no solution as, by a famous result of \textit{A. S. Besicovitch} [Math. Z. 27, 312--320 (1927; JFM 53.0713.01)], there exist Kakeya sets of measure zero. Nonetheless, variants of Kakeya's problem sparked important mathematical research for a period of almost 100 years. Motivated by recent result on Kakeya sets over finite fields by \textit{Z. Dvir} [J. Am. Math. Soc. 22, No. 4, 1093--1097 (2009; Zbl 1202.52021)], this article investigates Kakeya-like problems in an algebraic context for subsets of affine space \(\mathbb{A}^n\) over an arbitrary algebraically closed field. Call a subset ``Kakeya'' if it contains a line in every direction. The author considers the extreme case, where the Kakeya set is the union of finitely many locally closed sets and an open set whose complement is an irreducible hypersurface \(V(g)\) of degree \(d\). He shows that for every degree \(d\), there exist Kakeya sets in \(\mathbb{A}^3\) that are composed of the complement of \(V(g)\) and just two further points. If the complement of an irreducible hypersurface \(V(g) \subset \mathbb{A}^3\) can be made Kakeya by adding finitely many points, the ideal curve of \(V(g)\) has to satisfy some rather particular conditions. Call an \(n\)-dimensional subvariety \(X\) of \(\mathbb{A}^N\) ``Kakeya'' if the dimension of the set of all directions of lines contained in \(X\) equals \(n-1\). Examples of Kakeya subvarieties in this sense are hypersurfaces of degree \(d \leq N-1\). The author constructs a morphism \(\pi: X \to \mathbb{A}^n\) that maps the lines in a Kakeya subvariety \(X\) to lines in \(\mathbb{A}^n\) whose directions contain an open set.
scientific article; zbMATH DE number 6363064
scientific article