12 problems

Let $A\subseteq \mathrm{SO}(3,\mathbb{R})$ be open, and let $\mu$ denote normalized Haar measure on $\mathrm{SO}(3,\mathbb{R})$. For a unit vector $u\in\mathbb{R}^3$, define the an…

Let $d\geq 2$ be fixed, let $q$ be a sufficiently large prime power with $q\equiv 1\pmod d$, and let $G\subseteq {\mathbb F}_q^*$ be the multiplicative subgroup of index $d$. For s…

Continuous Babai conjecture. There is an integer $m$ such that

Breuillard–Green conjecture. The minimal value of the doubling constant for subsets of small measure should be about

Multiplicative bounded-product conjecture. There exist an infinite set $B\subset\mathbb{N}$ and $t\in\mathbb{N}$ such that

Elekes–Ruzsa conjecture. If

Let $G$ be a non-abelian finite simple group, and let $S_1,\ldots,S_k$ be normal subsets of $G$, meaning unions of conjugacy classes. Gill–Pyber–Szabó's product conjecture. There e…

Let $d$ be a positive integer, let $RY=\{ry:r\in R,\ y\in Y\}$ for $R\subseteq(0,\infty)$ and $Y\subseteq\mathbb{R}^d$. Dual fractal-set conjecture. For every $\alpha\in[0,1]$, the…

Let $d$ be a positive integer, let $\beta\beta\in[0,d]$, and let $RY=\{ry:r\in R,\ y\in Y\}$ for $R\subseteq(0,\infty)$ and $Y\subseteq\mathbb{R}^d$. Fractal-set conjecture. For ev…

Let $G$ be a torsion-free group, let $C$ be a finite subset of $G$ containing the identity element, and let $n$ be a positive integer. For a finite subset $X\subseteq G$, write…

Sárközy–Stewart conjecture. For every $\epsilon$ satisfying $0<\epsilon<1$, there exist $N(\epsilon)$ and $C(\epsilon)>0$ such that, for every integer $N\geq N(\epsilon)$ and every…

Finite-field product-set conjecture. There is an absolute constant $c>0$ such that, whenever