Density regions, integer certificates and packing colorings of distance graphs

Abstract: We study simultaneous color densities in packing colorings of integer distance graphs. For $D(1,6)$, we determine several exact density regions and prove that colors $1$ through $7$ have maximum combined density $211/252$. When this maximum is approached, the seven individual color frequencies are forced to converge to a specified vector. On an optimal low-color layer, some density vectors have nonperiodic realizations but no periodic realization; we determine how much accumulated density loss is necessary for switching between the relevant configurations. For sufficiently large additional color indices, a fixed finite graph describes the joint density region. In particular, we determine a seven-vertex region for every $i\equiv8\pmod{14}$ with $i\ge36$ and prove that $36$ is the first stable index in this residue class. The proofs combine finite-state integer certificates with explicit constructions and limit arguments. Applications give $17\leχ<em>ρ(D(1,6))\le20$, $18\leχ</em>ρ(D(1,8))\le22$, and $χ_ρ(D(1,9))\le17$.