19 problems

Let $Y\subset\mathbb{R}^n$ be Borel, and for $x\in\mathbb{R}^n$ let $\pi_x(Y)$ denote the radial projection of $Y\setminus\{x\}$ from $x$ onto the unit sphere. Liu's conjecture. ……

Furstenberg's countability conjecture. The set $E(K)$ should be no more than countable.

Let $T_m=m(m+1)/2$ and let $E(8T_x+8T_y+T_z)$ be the set of positive integers not represented by $8T_x+8T_y+T_z$. Since $8T_x+1$ is an odd square, a sum of two odd squares and a tr…

Let $A\subset\mathbb{R}^2$ be Borel. Let $\pi_0(A\setminus\{0\})$ be the radial projection from the origin, and let $E_+(A)$ denote the corresponding positive-dimensional exception…

Cholak–et al. sharpness conjecture. The lower bounds obtained for $T_{n,k}(a,s)$ by Cholak et al. are sharp for all $k$. The source does not reproduce the proposed lower-bound form…

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

Barron's dimension conjecture. The lower bound for the Hausdorff dimension of $L_\alpha$ should be

Let $X$ be a geometrically rationally connected, geometrically integral variety over a number field $F$, with $L$ a big and nef divisor. Let $Z_0$ be the initial exceptional subset…

Let $X$ be a smooth projective geometrically integral Fano variety over a number field, and set $L=-K_X$. For a thin map $f:Y\to X$ from a smooth projective geometrically integral…

Let $K\subset\mathbb{R}^{2}$ be a Borel set, and let $\pi_e\colon\mathbb{R}^{2}\to\mathbb{R}$ be the orthogonal projection $\pi_e(x)=x\cdot e$ for $e\in S^{1}$. Assume that…

Let $\Lambda=(\lambda_n)$ be the frequency sequence and let $L$ and $L^{-}$ denote the classes of auxiliary functions used in the paper. For $\varphi\in L$, define … Let $h$ be the…

Exceptional-set dimension conjecture. There exists a constant $C>0$ such that, for every such $x_0$ and $r$,

Let ${\mathcal A},{\mathcal B}\subseteq {\mathbb N}$. Write $\overline{{\mathcal A}+{\mathcal B}}$ for the exceptional set associated with the sumset ${\mathcal A}+{\mathcal B}$, l…

Let … where $T_z=z(z+1)/2$, and let $E(f)$ be the set of positive integers not represented by $f$. Sun's conjecture. The exceptional set is … The statement is presented as a comput…

For the four mixed forms … let $E(f)$ denote the set of positive integers not represented by $f$. Sun's conjecture. The exceptional sets are … … The source says that Kane confirmed…

Let $E\subset\mathbb{R}^2$ have Hausdorff dimension $\beta\leq 1$. For $\omega\in S^1$, let $p_\omega(E)$ be the orthogonal projection of $E$ onto the line through the origin in th…