Arithmetic limiting laws over shifted primes: dimension zero, Fourier decay, and absolute continuity

Abstract

For each prime $p$, let $c(p)=\frac{\phi (p-1)}{p-1}{\prod }_{j\ge 1}(1-p^{-j})$, the limiting probability, as $n\to \infty$, that a uniformly chosen matrix in $M_n({\mathbb{F}}_p)$ has primitive-root determinant. As $p$ ranges over the primes, $c(p)$ has the classical shifted-totient limiting law ${\mu }_G$ on $[0,1/2]$. Let ${\mu }_f$ be its image under $x\mapsto -\log x$. We prove that ${\mu }_G$ has Hausdorff dimension zero and gives full mass to absolutely normal Liouville numbers. Every Borel set of positive ${\mu }_G$-mass has packing dimension at least $1/2$, and local dimension fails to exist ${\mu }_G$-almost everywhere. Both Fourier transforms satisfy the effective bound $O((\log |\tau |{)}^{-1/3}(\log \log |\tau |{)}^{1/3})$. In every fixed reduced progression, the empirical laws of $c(p)$ and $\phi (p-1)/(p-1)$ approach their respective limiting laws with sharp Kolmogorov order $1/\log x$. Over unrestricted primes, these laws and their negative-logarithmic images converge in $W_1$ at rate $O_B((\log x{)}^{-B})$ for every fixed $B>0$. We obtain concentration expansions to every fixed inverse-logarithmic order and prove that every sufficiently short maximizing interval for ${\mu }_f$ contains $\log 3$. For $U_T$ uniform on $[0,T]$, $(\log T)\widehat{{\mu }_f}(U_T)$ converges in distribution to a circularly symmetric random variable whose modulus is an explicit random Euler product, with convergence of absolute moments of every real order $s>-1$. For every $A>0$, the strongly additive function $f_A$ with prime values $(\log \ell {)}^{-A}$ has absolutely continuous limiting laws for $f_A(n)$ and $f_A(p-1)$. We give extremal estimates and certified NTT prime searches. A lattice supplement determines second distinct nonzero squared norms of type-$A$ tensor products and gives uniform cyclotomic smoothing asymptotics at $\epsilon =2^{-c\phi (m)}$ for every fixed $c>0$. These statements are unconditional.

arXiv commentSubstantially expanded from the last version: proves absolute continuity for all positive logarithmic-jump exponents, giving counterexamples to the 1979 Erdos-Katai singularity conjecture; adds sharp prime-sampling rates, raw-ratio Fourier decay and normality, local-dimension oscillation, and complex Fourier-amplitude laws. Uniform cyclotomic smoothing is extended to every positive error exponent