3 problems

Let $A\subseteq\mathbb{F}_p^3$ be a set with $|A|=p$. A determined direction is a direction of a line passing through at least two points of $A$. Directional dichotomy for sets of…

Let $S\subseteq\mathbb{F}_p^3$ be a set of $p^2$ points, where $p$ is prime. The cylinder condition means that $S$ is the union of $p$ pairwise disjoint parallel lines, and hence h…

Let $E\subset{\Bbb R}^d$, with $d\geq 2$, have Hausdorff dimension $d-1$. Suppose that $E$ is rectifiable and is not contained in a hyperplane. Rectifiable-set conjecture. Then the…