[Paper Review] On Gaps Between Primitive Roots in the Hamming Metric

Rainer Dietmann, Christian Elsholtz+1 more|arXiv (Cornell University)|Jul 3, 2012

Analytic Number Theory Research22 references4 citations

TL;DR

This paper studies the Hamming distance between integers and primitive roots modulo a prime $p$, showing that any integer $n \in [1,p]$ can be transformed into a primitive root by flipping at most $0.11002786\ldots r$ bits, where $r$ is the number of bits in $p$. This improves upon the classical Burgess bound and establishes tighter bounds on the sparsity and distribution of primitive roots and quadratic non-residues in the Hamming metric.

ABSTRACT

We consider a modification of the classical number theoretic question about the gaps between consecutive primitive roots modulo a prime $p$, which by the well-known result of Burgess are known to be at most $p^{1/4+o(1)}$. Here we measure the distance in the Hamming metric and show that if $p$ is a sufficiently large $r$-bit prime, then for any integer $n \in [1,p]$ one can obtain a primitive root modulo $p$ by changing at most $0.11002786...r$ binary digits of $n$. This is stronger than what can be deduced from the Burgess result. Experimentally, the number of necessary bit changes is very small. We also show that each Hilbert cube contained in the complement of the primitive roots modulo $p$ has dimension at most $O(p^{1/5+ε})$, improving on previous results of this kind.

Motivation & Objective

• Investigate the minimal number of bit flips required to transform any integer into a primitive root modulo a prime $p$ under the Hamming metric.
• Improve upon the classical Burgess bound of $p^{1/4+o(1)}$ for gaps between primitive roots by using Hamming distance instead of additive distance.
• Analyze the sparsity of primitive roots and quadratic non-residues by bounding their minimal Hamming weight.
• Study the maximal dimension of Hilbert cubes contained entirely in the complement of primitive roots or quadratic non-residues modulo $p$.
• Explore the implications of these bounds for the distribution and density of such number-theoretic sets in binary space.

Proposed method

• Use character sum estimates over integers within a Hamming ball of radius $s$ around a given $n$ to analyze the density of primitive roots.
• Apply results from [22] and [1] to bound exponential sums over Hamming neighborhoods, leveraging the binary entropy function $H(\gamma)$.
• Define $\rho_0$ as the unique solution to $H(\rho) = 1/2$ in $[0, 1/2]$, which determines the optimal bit-flip threshold for reaching a primitive root.
• Use the binary entropy function $H(\gamma) = \frac{-\gamma\log\gamma - (1-\gamma)\log(1-\gamma)}{\log 2}$ to model the size of Hamming balls.
• Apply a recent result from [2] to refine the bound on the minimal Hamming weight $w_p$ of quadratic non-residues.
• Analyze Hilbert cubes $\mathcal{H}(a_0; a_1, \dots, a_d)$ in $\mathbb{F}_p$ to bound the maximal dimension $F(p)$ of cubes avoiding primitive roots.

Experimental results

Research questions

• RQ1 Can the bound $\Delta_p \leq (\rho_0 + o(1))r$ be improved under the Generalized Riemann Hypothesis?
• RQ2 Does the asymptotic density $f_i(x) = \frac{1}{\pi(x)}\#\{p \leq x : \Delta_p = i\}$ converge as $x \to \infty$?
• RQ3 Is $\Delta_p$ bounded for all primes $p$, or does it grow with $p$?
• RQ4 Does $w_p \leq 2$ hold for most primes, i.e., does $\lim_{x\to\infty} \frac{1}{\pi(x)} \sum_{p\leq x} w_p = \frac{3}{2}$?

Key findings

• The paper proves $\Delta_p \leq (\rho_0 + o(1))r$ as $p \to \infty$, where $\rho_0 \approx 0.11002786$ is the unique root of $H(\rho) = 1/2$, significantly improving the classical Burgess bound of $p^{1/4+o(1)}$.
• For the minimal Hamming weight $w_p$ of quadratic non-residues, the bound $w_p \leq (\vartheta_0 + o(1))r$ is established with $\vartheta_0 = \frac{1}{8\sqrt{e}} \approx 0.07581633$.
• The maximal dimension $F(p)$ of a Hilbert cube avoiding primitive roots modulo $p$ satisfies $F(p) \leq p^{1/5 + o(1)}$, improving on the previous bound of $12p^{1/4}$.
• Computational evidence shows that $\Delta_p = 2$ for most primes $p \leq 10^6$, and only 24 primes $p \leq 3 \times 10^6$ have $\Delta_p = 3$, with none having $\Delta_p \geq 4$.
• The proportion of primes with $w_p = 1$ is approximately $1/2$, and with $W_p = 1$ (i.e., $2$ is a primitive root) is approximately the Artin constant $A \approx 0.3739558$, confirming theoretical expectations.
• Under the GRH, the bound on $w_p$ can be improved to $O(\log r)$, and $W_p$ can be bounded by $O(\log r)$, suggesting efficient constructions of sets guaranteed to contain primitive roots.