Denominator-preserving maps

TL;DR

This paper proves that continuous injective maps between open subsets of R^n that bijectively map rational points with infinitely many denominators preserve Lebesgue measure, revealing a measure-preserving property under specific rational point conditions.

Contribution

It establishes a new link between rational point mappings and measure preservation for a class of continuous injective maps in Euclidean space.

Findings

Such maps preserve Lebesgue measure under the given conditions

Rational point bijections with infinitely many denominators imply measure preservation

The result connects rational structure with geometric measure theory

Abstract

Let F be a continuous injective map from an open subset of R^n to R^n. Assume that, for infinitely many k>1, F induces a bijection between the rational points of denominator k in the domain and those in the image (the denominator of (a_1/b_1,...,a_n/b_n) being the l.c.m. of b_1,...,b_n). Then F preserves the Lebesgue measure.