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

Vipin Singh Sehrawat

cryptography

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

arXiv:2603.11196

summary

The paper studies the limiting fraction of n×n matrices over finite fields with primitive‑root determinants, derives precise asymptotics and Fourier‑decay rates for associated singular measures, and applies these results to lattice‑smoothing bounds and cryptographic constructions such as Ring‑LWE and NTT‑friendly prime generation.

Abstract

For a prime $p$ , let $c (p) = \frac{φ ( p - 1 )}{p - 1} \prod_{j \geq 1} (1 - p^{- j})$ , the limiting density of matrices over $F_{p}$ whose determinant is a primitive root. We determine its limiting law over the primes: a continuous prime-indexed Bernoulli product supported on $[0, \frac{1}{2}]$ . The limiting measure has Hausdorff dimension zero and vanishing lower and upper dyadic $L^{q}$ dimensions for every $q > 1$ . Its logarithmic push-forward $μ_{f}$ is nevertheless Rajchman, unconditionally. For every $A > 0$ , as $T \to \infty$ , one has $∣ μ_{f} (τ) ∣ \leq (lo g T)^{- 1 + o (1)}$ outside a subset of $[0, T]$ of relative measure $O_{A} ((lo g lo g T)^{- A})$ . Its concentration function satisfies $sup_{a} μ_{f} ([a, a + h]) \sim S_{2} e^{- γ} / lo g (1/ h)$ , and every maximizing left endpoint lies near $lo g 3$ . We also prove $min_{p \leq x} c (p) ≍ (lo g lo g x)^{- 1}$ and $lim sup_{p \to \infty} (c (p) lo g lo g p)^{- 1} = e^{γ}$ . The limiting law of $lo g (φ (p + 1) / φ (p - 1))$ has full support, is purely singular, and has Hausdorff dimension zero. We further prove a shifted-prime analogue of Gronwall's theorem and derive explicit bounds for $1/ c (p)$ without complete factorization of $p - 1$ , yielding a certified asymptotic search for fully splitting NTT primes. Under an explicit unproved hypothesis on exponent-pair constants, $∣ μ_{f} (τ) ∣ = O ((lo g lo g ∣ τ ∣)^{- 1/2})$ . Finally, exact shell gaps of cyclotomic codifferents yield a uniform smoothing asymptotic at $ε = 2^{- c φ (m)}$ for $c > 2 lo g_{2} (1 + 6)$ .

Substantially revised and reorganized. The qualitative Rajchman theorem remains unconditional; the quantitative pointwise Fourier-decay estimate is now conditional on an explicit hypothesis. Added sharp modulus-of-continuity and typical-frequency results; removed the general lattice-smoothing section

Topics & keywords

#primitive-root determinants #finite field matrices #fourier decay #rajchman measures #lattice smoothing #ring-lwec(p)primitive-root determinant densityFourier decay rateRajchman transformlattice smoothing boundNTT-friendly primes