12 problems

Let $P_t:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be an analytic family of linear maps. Define $f:[0,n]\rightarrow\mathbb{R}$ as follows: for an integer $x\in[0,n]$, let $f(x)$ be the…

Let $P_t:\mathbb{R}^n\rightarrow\mathbb{R}^m$ be an analytic family of linear maps, meaning that its coordinate functions are analytic functions of $t$. Assume that for every subsp…

Let $K$ be a self-similar set in the plane, and call a direction $\pi\in G(2,1)$ exceptional for orthogonal projections of $K$ if the projection dimension equality does not hold. C…

Let $E\subset\mathbb{R}^d$ and let $\mathcal{V}\subset G(d,d-1)$ be Borel sets such that $\dim_{\mathcal{H}}\mathcal{V}>0$ and $\mathcal{V}^{\perp}$ is not contained in a great cir…

Let $A\subset\mathbb{F}_p^3$ satisfy … Recall the exceptional set $E_s(A)$ for projection to lines or planes. Let $t(a,s)$ be the lower bound for $T(a,s)$ given by Proposition 1 fo…

Let $T(a,s)$ denote the exponent governing the exceptional-set estimate, and let the lower bounds for $T(a,s)$ be those established in Propositions 1 and 2. Sharp bounds conjecture…

Let $A\subset \mathbb{F}_p^2$ satisfy … For $0<s<\max\{1,a\}$, let $E_s(A)$ be the exceptional set defined for projection to lines, with $n=1$ and $k=1$. The finite-field exception…

The exceptional-set projection conjecture. Then

Let $s\in (0,1)$ and $C_0\in (1,\infty)$. Let $E\subset \mathbb{R}^2$ be a bounded AD-regular set with constant $C_0$, meaning that $E$ is closed and, for every $x\in E$ and…

Let $E\subset\mathbb{R}^2$ be $\mathcal{H}^1$ measurable and purely $1$-unrectifiable, with $\mathcal{H}^1(E)<\infty$. Besicovitch's almost-everywhere line intersection conjecture.…

Let $J\subset\mathbb{R}$ be a bounded open interval, let $\gamma\colon J\to S^2$ be a $\mathcal{C}^2$ curve satisfying … and let $\rho_\theta\colon\mathbb{R}^3\to\mathbb{R}$ be the…

Let $T_{m,n}:\mathbb{T}^2\to\mathbb{T}^2$ denote the coordinatewise action of $T_m$ and $T_n$, let $\Pi_{2,1}$ be the set of orthogonal projections from $\mathbb{R}^2$ to…