The shifted-prime Erdős-Wintner law for primitive-root determinant densities: extremal order, dimension zero, and Fourier decay

Abstract

For a prime $p$, let $c(p)=\frac{φ(p-1)}{p-1}\prod_{j\ge1}(1-p^{-j})$, the limiting density of matrices over $\mathbb F_p$ whose determinant is a primitive root. Its limiting law over the primes is a continuous prime-indexed Bernoulli product with support $[0,1/2]$, Hausdorff dimension zero, and vanishing lower and upper dyadic $L^q$ dimensions for every $q>1$. Its logarithmic push-forward $μ_f$ is Rajchman. For every $A>0$, as $T\to\infty$, $|\widehat{μ_f}(τ)|\le(\log T)^{-1+o(1)}$ outside a subset of $[0,T]$ of relative measure $O_A((\log\log T)^{-A})$. As $h\downarrow0$, $\sup_aμ_f([a,a+h])=\mathfrak S_2e^{-γ}/\log(1/h)+O(\log^{-2}(1/h))$, where $\mathfrak S_2$ is the twin-prime singular series; every maximizing left endpoint lies in $(\log3-h,\log3+h)$ for sufficiently small $h$. We prove $\min_{p\le x}c(p)\asymp(\log\log x)^{-1}$ and $\limsup_{p\to\infty}(c(p)\log\log p)^{-1}=e^γ$. The limiting law of $\log(φ(p+1)/φ(p-1))$ has support $\mathbb R$, is purely singular, and has Hausdorff dimension zero. These results are unconditional. We give an effective refinement of the shifted-prime $σ$-extremal order, with explicit high-prime-power moduli, least-prime witnesses, and an effective one-sided lower bound. Bounds for $1/c(p)$ without complete factorization of $p-1$ yield a certified asymptotic search for fully splitting number-theoretic transform (NTT) primes with a prescribed reciprocal-density bound at fixed transform length. Under an explicit unproved hypothesis on exponent-pair constants, $|\widehat{μ_f}(τ)|=O(1/\log\log|τ|)$. Finally, we determine the second distinct squared norm of $A_{n_1}\otimes\cdots\otimes A_{n_k}$ for $k\ge2$ and $n_i\ge2$. This yields exact shell gaps of cyclotomic codifferents and a uniform smoothing asymptotic at $ε=2^{-cφ(m)}$ for $c>2\log_2(1+\sqrt6)$.