Primitive-Root Ratio over Prime Fields: A Shifted-Prime Distribution of Hausdorff Dimension Zero and Implications for PRIM-LWE

For a prime $p$, let $c(p)$ denote the limiting fraction of $n\times n$ matrices over ${\mathbb{F}}_p$ whose determinant is a primitive root modulo $p$. The quantity $c(p)$ is a natural multiplicative deformation of the totient ratio $\phi (p-1)\mathrm{/}(p-1)$ and inherits its distributional behaviour over the primes. Existence and continuity of the limiting law follow from the shifted-prime Erdős–Wintner–Hildebrand framework. We prove the following new results: unconditionally, ${\inf }_{p}c(p)=0$ and the sharp order is ${\min }_{p\le x}c(p)\asymp 1\mathrm{/}\log \log x$; the reciprocal satisfies ${\mathrm{lim\hspace{0.17em}sup}}_{p\to \mathrm{\infty },p\text{ prime}}1\mathrm{/}(c(p)\log \log p)=e^{\gamma }$, and no smaller constant suffices. We give a complete proof, combining an adaptation of Erdős’s argument with the Jessen–Wintner pure-type dichotomy, that the limiting distribution is purely singular, and strengthen this to ${\dim }_H({\mu }_G)=0$, i.e. the limiting measure is carried by a Borel set of Hausdorff dimension zero. The distribution has full topological support $[0,\frac{1}{2}]$ and admits a Bernoulli-product representation indexed by the odd primes. The moments are given by convergent Euler products, and the Mellin transform $\mathbb{E}[X^s]$ extends to an entire function of $s$, non-vanishing on $\mathrm{Re}(s)>0$. Near the right endpoint, $1-G(\frac{1}{2}-\epsilon )\sim \kappa \mathrm{/}\log (1\mathrm{/}\epsilon )$ with an explicit constant $\kappa$. The quantity $1\mathrm{/}c(p)$ equals the dimension-uniform expected rejection-sampling overhead in the reduction from learning with errors (LWE) to PRIM-LWE in lattice-based cryptography. The explicit bounds yield concrete overhead estimates for all primes appearing in current NIST post-quantum standards and representative NTT-friendly moduli.