7 problems

Let $E\subset\mathbb{R}^{2}$ be a $1$-set, meaning that $E$ has locally finite $\mathcal{H}^{1}$ measure, and let … Define $\sigma_{1,2}$ as the greatest value that the lower $1$-d…

Let $E\subset\mathbb{R}^2$ be $\mathcal{H}^1$ measurable, with $\mathcal{H}^1(E)<\infty$, and suppose that $E$ is purely $1$-unrectifiable. Besicovitch's one-half density conjectur…

Let ${\mathcal{F}}$ be a convergent $F$-isocrystal on the curve $U$, fix $u\in U({\mathbb{F}}_{q^e})$, and let $\operatorname{Gr}({\mathcal{F}}/U,u)$ be its monodromy group over…

Let $U$ be the curve, let $K$ be the specified finite totally ramified extension, and let ${\mathcal{F}}$ be a convergent $F$-isocrystal on $U$. Fix $u\in U({\mathbb{F}}_{q^e})$, a…

Let $0<\alpha<1$ and let $\epsilon>0$. For a positive integer $N$, write $[N]=\{1,2,\dots,N\}$. A corner in $[N]^2$ is a triple $((x,y),(x+d,y),(x,y+d))$ with integer $d\neq0$; its…

Let $0<\alpha<1$, let $\epsilon>0$, and let $k=3$ or $4$. For a positive integer $N$, write $[N]=\{1,2,\dots,N\}$. A length-$k$ arithmetic progression in $[N]$ has a common differe…

Szenes's conjecture. The constant $\delta_{\mathcal{H}}$ equals Szenes's upper bound, namely the root in the stated range of