6 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…

For $delta\bin(0,1)$, define the Nikodým maximal operator by … where the supremum is over all $delta$-tubes $T$ centered at $x$, and a $delta$-tube is the closed $(delta/2)$-neighb…

Córdoba's maximal Nikodym conjecture. For every $\epsilon>0$, there is a constant $C_\epsilon$ such that

Let $d$ be a positive integer and let $q$ be the order of a finite field. Write $\operatorname{Nikodym}(d,q)$ for the minimum size of a Nikodym set in the $d$-dimensional affine sp…

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…

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…