Optimal Hypercontractivity and Log--Sobolev inequalities on Cyclic Groups $\mathbb{Z}_{m\cdot 2^k}$
Gan Yao
TL;DR
The paper advances the sharp hypercontractivity theory for Poisson-like semigroups on cyclic groups Z_n by recasting the Log–Sobolev inequality (LSI) in Fourier form and solving it via a Karush–Kuhn–Tucker (KKT) analysis. It introduces tighter base LSIs on Z_4 and Z_6 using modified weights, and then carries out a dyadic induction to lift LSIs to larger groups through a Cooley–Tukey Dirichlet-form comparison. The main achievement is proving optimal LSI constant 2 and the corresponding hypercontractivity time t_pq = (1/2) log((q-1)/(p-1)) for n in {3·2^k, 2^k}, by constructing an induction framework that propagates these inequalities along the towers n, 2n, 4n, etc. The work highlights a pathway toward understanding LSIs on broader towers (m·n^k) and leaves open the general n case, which demands further structural insights.
Abstract
For $1<p\le q<\infty$ 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 $ψ_n(k)=\min(k,n-k)$, is hypercontractive from $L_p$ to $L_q$ if and only if $t\ge \tfrac{1}{2}\log\big(\tfrac{q-1}{p-1}\big)$. 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.