Furstenberg sets and Furstenberg schemes over finite fields (Q318929)

In the paper under review, the authors investigate discrete and finite-field analogues of the \(k\)-plane Furstenberg set problem which is concerning with the Hausdorff dimension of a compact subset \(S\subseteq {\mathbb R}^n\) provided that for every \(k\)-plane \(W\subseteq {\mathbb R}^n\) there is a \(k\)-plane parallel to \(W\) such that \(W\cap S\) has Hausdorff dimension at least \(c\); [\textit{T. Wolff}, in: Prospects in mathematics. Invited talks on the occasion of the 250th anniversary of Princeton University. Papers from the conference, Princeton, NJ, USA, March 17--21, 1996. Providence, RI: American Mathematical Society. 129--162 (1999; Zbl 0934.42014)].NEWLINENEWLINEThroughout for \(X\) a set, \(|X|\) denotes its number of elements, and \(|X|\succ f(q,n,k,c)\) means \(|X|>Cf(q,n,k,c)\) with \(C\) being a constant which is independent of \(q\). Let \(S\subseteq {\mathbb F}^n\), where \(\mathbb F\) is the finite field of order \(q\). Suppose that for every \(k\)-plane \(W\subseteq {\mathbb F}^n\) there exists a \(k\)-plane \(W'\) parallel to \(W\) such that \( |W'\cap S|\geq q^c\), \(c\in[0,k]\). What can be said about \(|S|\)? One of the main result here is the fact that \(|S|\succ q^{cn/k}\) (cf. [J. Am. Math. Soc. 22, 1093--1097 (2009; Zbl 1202.52021)]). This result can be extended to \(0\)-dimensional subschemes of the affine space \(\mathbb A^n\) over \(\mathbb F\), where the proof is based on well-known degeneration algebraic geometric techniques.