5 problems

Let $d\geq 3$, let $\mathbb{F}_q^d$ be the $d$-dimensional vector space over the finite field with $q$ elements, and let a weak Nikodym set be a subset of $\mathbb{F}_q^d$ containi…

Let $(M^d,g)$ be a Riemannian manifold of dimension $d$ and constant sectional curvature, and let a Nikodym set be a Borel set whose associated Nikodym enlargement has positive mea…

For $1\leq d<n$, let $\mathcal N_\delta$ denote the $(d,n)$-Nikodym maximal function. For $1\leq p\leq n-d+1$, set $q=p'(n-d)$, where $p'$ is the Hölder conjugate of $p$. Off-diago…

Nikodym maximal-function conjecture. For $1<p<n-d+1$,

Let $L$ be a set of lines in $\mathbb{F}_q^3$, and define the union of lines by … Weak union-of-lines conjecture. If … and no plane contains $\omega(q)$ lines of $L$, then … This c…