3 problems

Let $G$ be a definably compact group, let $G^{00}$ be its type-definable connected component, and let $\pi\colon G\to G/G^{00}$ be the quotient map. Write $m$ for Haar measure on t…

Let $G$ be a compact simple group of rank $r$, let \textit{\text{\boldmathvarphi}} denote angular variables on a maximal Abelian torus, and let \textit{\text{\boldmathchi…

Let $G$ be a profinite group, and let $X_n(G)$ denote the set of solutions of $x^n=1$ in $G$. Write ${\mathbf m}_G$ for the normalized Haar measure on $G$. Lé vai–Pyber conjecture.…