14 problems

Massicot–Wagner's conjecture. Even without the definable amenability assumption, a suitable Lie model exists for approximate subgroups.

Let $\mathbb F$ be an algebraically closed field, let $d\geq1$, and let $A\subset\operatorname{GL}_d(\mathbb F)$ be a $K$-approximate group. Suppose every $a\in A^2$ has an eigensp…

Let $\mathbb F$ be a field, let $d\geq1$, and let $A\subset\operatorname{GL}_d(\mathbb F)$ be a $K$-approximate group for some $K\geq1$. Polynomial-dimension abelian-structure conj…

Let $G$ be a group and let $A\subset G$ be a finite symmetric set satisfying … A subset is commuting if all of its elements commute pairwise. The general abelian-structure conjectu…

Small tripling conjecture. For every $K\ge 1$ and $\varepsilon>0$, this implies

Let $q=p^r$ be any prime power, and let ${\mathbb{F}}_q$ be the corresponding finite field. Consider the Gill–Helfgott result for subsets $A\subseteq GL_n({\mathbb{F}}_p)$ whose ge…

Let $G$ be a group acting on a set $X$, let $A$ be a subset of $G$, and let $Y$ be a finite subset of $X$. Write $A(Y)$ for the image of $Y$ under $A$, and let $K\geq 1$ satisfy…

Large type-definable subgroup conjecture. Every infinite approximate subgroup contains a large type-definable subgroup.

Additive volume conjecture. The source statement is incomplete in the supplied excerpt: it begins “we have” but does not provide the asserted conclusion. The conjecture therefore c…

Lie-model conjecture. Even without the definable amenability assumption, every suitable approximate subgroup admits a suitable Lie model.

Let $S$ be a generating set in the setting of the theorem above, whose conclusion gives a bound involving a constant $C_d$ and an implied multiplicative constant, with $d$ denoting…

Let $\alpha$ be a symmetric subset of $GL(n,\mathbb{F}_p)$ containing $1$ and satisfying … for some $K\ge 1$. Folklore conjecture. There are subgroups $S\triangleright P$ of…

Let $A\subseteq \operatorname{SL}_n(\mathbb{C})$ be a $K$-approximate group, and say that a set $B$ $K'$-controls $A$ when the source's approximate-group control relation holds. Hr…

Let $G$ be a locally compact group, let $A\subset G$ be a compact neighborhood, and let $\mu$ be Haar measure on $G$. For $n\geqslant 1$, write $A^n$ for the $n$-fold product set o…