5 problems

Let $S\subseteq\mathbb{N}$ be a simple set, meaning that $\mathbb{N}\setminus S$ is infinite and contains no infinite recursively enumerable subset. Let $U\subseteq\mathbb{Q}$ be t…

Let $P$ be the set of all primes and let $WQ e P$ be a subset of $P$. Define … For a variety $V$ over $\mathbb{Q}$, write $V(O_{Q,W_Q})$ for its points over this ring. Generalized…

Let $V$ be any variety over $\mathbb{Q}$, and let $V(\mathbb{Q})$ and $V(\mathbb{R})$ denote its rational and real points, respectively. Consider the topological closure of…

Let $\mathbb{Z}$ denote the ring of integers and $\mathbb{Q}$ the field of rational numbers. A subset of $\mathbb{Q}$ is Diophantine if it is defined by the existence of solutions…

Mazur's conjecture. The topological closure of $V(\mathbb{Q})$ in $V(\mathbb{R})$ possesses at most a finite number of connected components.