Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups ${\mathbb{Z}}_{m\cdot 2^k}$

For $1<p\le q<\mathrm{\infty }$ and $n\in \{3\cdot 2^k,2^k\}$ with $k\ge 1$, we prove that the Poisson-like semigroup $(P_t{)}_{t\in {\mathbb{R}}_{+}}$ on ${\mathbb{Z}}_n$, associated with the word length ${\psi }_n(k)=\min (k,n-k)$, is hypercontractive from $L_p$ to $L_q$ if and only if $t\ge \frac{1}{2}\log \left(\frac{q-1}{p-1}\right)$. We establish sharp Log–Sobolev inequalities with the optimal constant $2$, by performing a KKT analysis, and lifting from the base cases ${\mathbb{Z}}_6$ and ${\mathbb{Z}}_4$ via a Cooley–Tukey $n\mapsto 2n$ comparison of Dirichlet forms. The general case for arbitrary $n$ remains open.