Hilbert’s Irreducibility Theorem via Random Walks

Abstract Let $G$ be a connected linear algebraic group over number field $K$, let $\Gamma $ finitely generated Zariski dense subgroup of $G(K)$, and $Z\subseteq G(K)$ thin set, in the sense Serre. We prove that, if $G/\textrm {R}_{u}(G)$ is either trivial or semisimple $Z$ satisfies certain necessary conditions, then long random walk on Cayley graph hits elements with negligible probability. deduce corollaries to Galois covers, characteristic polynomials, fixed points actions. also analogous results case where $K$ global function field.