A Sieve on Rational Imbalances and the First Appearance of Denominators

TL;DR

The paper introduces the imbalance sieve, a new elementary method for enumerating rationals via the bilinear map $\delta(p,q)=\frac{p-q}{p+q}$ on pairs $(p,q)$ with $p\ge 2$ and $1\le q<p$, processed lexicographically. It proves that each reduced denominator appears exactly once, with the first appearance always yielding the unit fraction $\frac{1}{d}$, and it derives explicit quadratic index laws for odd and even denominators, ensuring a complete, nonrepeating enumeration of $(-1,1)\cap\mathbb{Q}$ that extends to all of $\mathbb{Q}$ via the Cayley transform. The work connects this sieve to classical rational enumerations, modular symmetry (PSL$(2,\mathbb{Z})$), and hyperbolic geometry, offering a transparent algebraic parametrization that complements mediant-based trees. Overall, the imbalance sieve reveals an unexpected, highly regular structure in rational enumeration with potential extensions to higher number fields and $p$-adic settings.

Abstract

We construct a sieve that enumerates rational ``imbalances'' of the form $(p-q)/(p+q)$ for integers $p\ge2$ and $1\le q<p$, ordered lexicographically by $(p,q)$. Each imbalance is reduced to lowest terms, and we record the sequence of distinct denominators as they first appear. We show that every positive integer occurs exactly once as such a denominator, and that its first appearance coincides with the unit fraction $1/d$. We then prove that the sieve, when viewed as a map from pairs $(p,q)$ to reduced fractions, enumerates all rational numbers in $(-1,1)$ without repetition, extend it symmetrically to all of $\mathbb{Q}$, and discuss its connections to hyperbolic geometry and rational enumeration theory.

Theorems & Definitions (9)

• Definition 2.1: Iteration index
• Definition 2.2: Reduced imbalance
• Lemma 3.1
• proof
• Corollary 3.2
• Definition 4.1
• Theorem 5.1: First Appearance of Denominators
• Lemma 7.1: Cayley transform
• Theorem 7.2: Extension to $\mathbb{Q}$