6 problems

Let a primitive integral Apollonian packing be an integral Apollonian packing whose curvature quadruples have greatest common divisor $1$. An integer occur in the packing when it i…

Let $A\subset\mathbb N$ be a finite alphabet and let $X_A$ be its limit set. For each $m\in\mathbb N$, define reduction modulo $m$ by … The alphabet $A$ has no local obstructions i…

Local Pre-Malle conjecture. There is a finite set of primes $S_0$ depending on $G$ such that, for every prime $p\notin S_0$ and every integer $k\not\equiv 0\pmod p$, some $G$-exten…

Local Pre-Malle conjecture. There is a finite set of primes $S_0$ depending on $G$ such that, for every prime $p\notin S_0$ and every integer $k\not\equiv 0\pmod p$, some $G$-exten…

Positive-density conjecture. The set $S(G)$ has positive density inside $\mathbb{N}_G$. This conjecture isolates the expected absence of widespread local obstructions to reduced di…

Anisotropic-prime exception conjecture. All but finitely many exceptions of a Type $B$ form are contained in $\mathcal{F}$.