A note on Buczolich’s solution of the Weil gradient problem: a construction based on an infinite game.

*(English)*Zbl 1127.26006In a recent paper [Rev. Mat. Iberoam. 21, 889–910 (2005; Zbl 1116.26007)] Z. Buczolich solved Weil’s gradient problem by establishing the existence of a differentiable function \( f : {\mathbb R}^2 \to {\mathbb R} \) such that \( {\nabla}f(0) = 0 \) and \( | {\nabla}f | \geq 1 \) almost everywhere. In this paper the authors give a simpler construction of such a function. For this purpose they consider the following infinite game.

Let \( B = B(0,R) \) be an open disk in \( {\mathbb R}^2 \). The point-line game PL is a sequence of rounds. The first and the second player choose points \( {\boldsymbol a}_k \) in \(B\) and lines \( p_k\), respectively, obeying the following rules. In the first round, the first player chooses a point \( {\boldsymbol a}_1 \in B \) and then the second player chooses a line \( p_1 \) with \( {\boldsymbol a}_1 \in p_1\). In the \(k\)-th round, where \( k > 1 \,\), the first player chooses a point \( {\boldsymbol a}_k \in B \cap p_{k-1} \) and the second player chooses a line \( p_k \) passing through \( {\boldsymbol a}_k \,\). The first player wins if the sequence \( ({\boldsymbol a}_k) \) diverges, otherwise the second player wins.

Two proofs are given for the crucial statement that the second player has a winning strategy in the game PL. Based on this result, a new proof of Buczolich’s theorem is presented in the paper.

Let \( B = B(0,R) \) be an open disk in \( {\mathbb R}^2 \). The point-line game PL is a sequence of rounds. The first and the second player choose points \( {\boldsymbol a}_k \) in \(B\) and lines \( p_k\), respectively, obeying the following rules. In the first round, the first player chooses a point \( {\boldsymbol a}_1 \in B \) and then the second player chooses a line \( p_1 \) with \( {\boldsymbol a}_1 \in p_1\). In the \(k\)-th round, where \( k > 1 \,\), the first player chooses a point \( {\boldsymbol a}_k \in B \cap p_{k-1} \) and the second player chooses a line \( p_k \) passing through \( {\boldsymbol a}_k \,\). The first player wins if the sequence \( ({\boldsymbol a}_k) \) diverges, otherwise the second player wins.

Two proofs are given for the crucial statement that the second player has a winning strategy in the game PL. Based on this result, a new proof of Buczolich’s theorem is presented in the paper.

Reviewer: Zoltán Boros (Debrecen)