Primitive-root determinant densities: extremal order, dimension-zero, Fourier decay, and a lattice-smoothing no-go

Abstract

$c(p)$: limiting fraction of $n\times n$ matrices over $\mathbb F_p$ with primitive-root determinant; $1/c(p)$: the rejection-sampling loss in PQ VSS; $c(p)=\tfrac{φ(p-1)}{p-1}\prod_{j\ge1}(1-p^{-j})$. Over primes, $c(p)$ follows $φ(p-1)/(p-1)$: continuous on $[0,1/2]$, $X=\tfrac12\prod_{\ell\ge3\,\mathrm{prime}}(1-1/\ell)^{B_\ell}$, $B_\ell$ independent, $\Pr(B_\ell{=}1)=1/(\ell{-}1)$. Hence $\inf_p c(p)=0$, $\min_{p\le x}c(p)\asymp1/\log\log x$, $\limsup_p 1/(c(p)\log\log p)=e^γ$ on a primorial progression. $μ_G$ is singular with $\dim_Hμ_G=0$, sharpening Erdős (1939) for $φ(n)/n$. Also $1-G(\tfrac12-ε)\sim\mathfrak S_2 e^{-γ}/\log(1/ε)$, $\mathfrak S_2$ the twin-prime singular series (infinitude not needed), and $μ_f=\logμ_G$ is Rajchman, with an explicit Fourier-decay rate under a Graham-Kolesnik envelope. $\mathbb E[X^{-1}]\approx2.83$ vs. worst case $(1+o(1))e^γ\log\log p$; a deterministic poly$(\log p)$-time, factoring-free two-sided certificate for $1/c(p)$ of gap $1+o(1)$; a Las Vegas generator of NTT-friendly primes $q\equiv1\pmod{2N}$; and $1/c(p)>2$ for all $p$, $1/c(p)\le(e^γ+o(1))\log y$ on $y$-friable shifts. The Micciancio-Regev smoothing $η_ε(Λ)$ has kissing floor $F=\sqrt{\ln(K/ε)/π}/λ_1(Λ^*)$ ($K$ the dual kissing number): at $ε=2^{-cn}$ every lattice has $F\leη_ε\le\sqrt{π/(\min(c,1)\ln2)}\,F$, while at fixed $ε$ the ratio $η_ε/F$ can diverge as $\sqrt{\log n}$. For cyclotomic $\mathbb Q(ζ_m)$ ((Ring-)LWE), an exact three-case law gives dual shell gap $g_m\in\{\sqrt{3/2},\sqrt2,\sqrt3\}$, infimum $\sqrt{3/2}$ on $\{m:ω_{\mathrm{odd}}(m)\ge2\}$, discharging the gap hypothesis unconditionally; at $ε=2^{-cφ(m)}$, $c>2\log_2(1+\sqrt6)$ pins $η_ε(Λ_m)$ to the floor within $1+O(1/(cφ(m)))$.