A Complete Solution of Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups ${\mathbb{Z}}_n$ for $n\ge 4$

For $1<p\le q<\mathrm{\infty }$ and $n\ge 4$, 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 $t\ge \frac{1}{2}\log \left(\frac{q-1}{p-1}\right)$. To this end, we establish the corresponding sharp Log–Sobolev inequalities with the optimal constant $2$.