10 problems

Let $0<\delta\ll1$, let $\Omega$ be a $\delta$-separated set of directions, and for each $\omega\in\Omega$ let $R_\omega$ be a $(\delta,1/2)_2$ set contained in a rectangle of dime…

Let $\delta\in(0,1)$. Let $(L,Y)_\delta$ be a set of directional $\delta$-separated lines in $\mathbb{R}^n$ with an $(\varepsilon_1,\varepsilon_2,C_Y)$-two-ends, $\lambda$-dense sh…

Let $E\subset\Omega$ be an $(s,t)$-Furstenberg set on a Riemannian surface: there is a family $\Gamma$ of local geodesics with $\dim_{\mathcal H}\Gamma\geq t$ such that…

Let $0<u\leq1$ and $0\leq v\leq3$. A sine wave $(u,v)$-Furstenberg set or circular $(u,v)$-Furstenberg set is a set $F\subseteq\mathbb{R}^2$ of the type specified in the paper, wit…

Let $\delta\in(0,1)$, and let $(L,Y)_\delta$ be a set of directionally $\delta$-separated lines in $\mathbb{R}^n$ with an $(\varepsilon_1,\varepsilon_2)$ two-ends, $\lambda$-dense…

Let $\mathfrak{f}(s,t)$ denote the dimension quantity defined in the paper, with $s$ and $t$ in the parameter ranges below. The conjecture is the dimension-function conjecture: for…

Fix $s\in(0,1]$ and $t\in(0,2]$. Let $E$ be an $(s,t;2,1)$-set over $\mathbb{F}_p$, meaning that $E$ is a union of subsets $Y(V)$ over a family of affine $1$-planes $V$ containing…

Let $0 \leq s \leq 1$. A Furstenberg $s$-set is a compact set $K \subset \mathbb{R}^{2}$ such that there is a set of unit vectors $S_K \subset S^1$ with positive length and, for ev…

Finite-field Furstenberg set conjecture. Then

Three-halves logarithmic dimension conjecture. The function