6 problems

Breuillard's growth conjecture. For every integer $d>0$, there is $c_d>0$ such that

Let $\mathcal{A}\subset\mathbb{R}$ be a non-empty finite set, and let $T(\mathcal{A})$ denote the quantity counting the relevant commuting pairs of $2\times 2$ matrices associated…

Let $S$ be a non-abelian finite simple group, and let $A,B$ be subsets of $S$. For $g\in S$, write $A^g=g^{-1}Ag$. Small-set asymmetric growth conjecture. For every $\epsilon>0$ th…

Let $S$ be a non-abelian finite simple group, and let $A,B$ be subsets of $S$. For $g\in S$, write $A^g=g^{-1}Ag$. Asymmetric growth conjecture. For every $\delta>0$ there exists…

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

Solvable linear-group growth conjecture. The stated alternative holds for every such $A$, field $K$, and parameter $C\geq 1$. This gives a structural explanation for failure of rap…