On the analytic bijections of the rationals in $[0,1]$

TL;DR

This paper investigates analytic functions on [0,1] that bijectively map rational points within the interval, revealing limited distributional constraints unless the functions possess algebraic properties, and explores related height-based questions.

Contribution

It provides an arithmetical analysis of such bijective analytic functions and examines the implications for the distribution of rational points and algebraic properties.

Findings

Existence of such functions shows limited distributional constraints.

Additional algebraic properties are needed to impose structure.

Height-related questions are also explored.

Abstract

We carry out an arithmetical study of analytic functions $f: [0,1] \to [0,1]$ that by restriction induce a bijection $\mathbb{Q} \cap [0,1] \to \mathbb{Q} \cap [0,1]$. The existence of such functions shows that, unless $f(x)$ has some additional property of an algebraic nature, very little can be said about the distribution of rational points on its graph. Some more refined questions involving heights are also explored.