3 problems

Let $\g\in\operatorname{GL}_2(\mathbb{Q})$ be a hyperbolic matrix, that is, one having two distinct real eigenvalues; equivalently … Let $\G=\langle\g\rangle^+:=\{\g^n:n\ge0\}$ be…

Local-global conjecture. If $\gd_{\cA}>1/2$, then $\cT_{\cA}$ contains every sufficiently large admissible integer. Moreover, for an admissible $t\in[N,2N)$,

Integral Apollonian packings assign integer curvatures to their circles, and an integer is called locally admissible when it satisfies the congruence obstructions arising from the…