14 problems

Let $E \subset \mathbb{R}^2$ be an $\alpha$-Furstenberg set, meaning that for each direction $e \in S^1$, there exists a line segment $\ell$ in direction $e$ such that…

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…

For $0<\beta\leq1$, a $\beta$-set is a compact set $K\subset\mathbb{R}^2$ such that for every direction $\omega\in S^1$ there is a line segment of direction $\omega$ intersecting…

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 $0<u\leq1$, and let $F\subseteq\mathbb{R}^2$ be a circular $(u,1)$-Furstenberg set. Liu's conjecture. One should have … This is the same lower bound as for linear $(u,1)$-Furst…

Let $t\in(0,2)$ and $n\geq4$. Let $|E|_\delta$ denote the minimum number of $\delta$-balls needed to cover a set $E\subset\mathbb{R}^n$. A family of $\delta$-tubes satisfies the…

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…

Zhang's sharpness conjecture. When $q$ is prime, this upper bound is sharp: the minimum possible size of such a set $S$ is of order $q^{(n+1)(c/2)+(n-1)/2}$.

Finite-field Furstenberg set conjecture. Then

Three-halves logarithmic dimension conjecture. The function

Dimension-function conjecture. If $E$ belongs to the class $F_{\mathfrak{h}\mathfrak{g}}$, then an appropriate dimension function for $E$ should be dimensionally greater than both

Limitation conjecture. This proof cannot be used to prove that an $F_{\mathfrak{h}}$-set has positive $\mathfrak{h}^2$ measure. In particular, when $\mathfrak{h}(x)=x$, such a conc…