Admissibility is defined for the integer $k$-tuples by the non-existence of a prime number $p$, for which each element of the $k$-tuple under union forms a complete residue system $\pmod{p}$.
If we define $H \subset \mathbb{Z}$ as a $k$-tuple and $H' = (\mathbb{Z}/m \# \mathbb{Z})^\times$, the multiplicative group of residue classes with a primorial modulus, such that the modulus $m\#$ is greater than the diameter of $H$.
Is it equivalent to say that if there exists a number $n \in \mathbb{Z}$, such that the translation on the number line $H \oplus n \subseteq H'$, that $H$ is admissible?