A short proof of a known result about the density of a certain set in $[0,1]^n$

TL;DR

This paper provides a simplified proof demonstrating the density of a specific set in the unit cube, extending previous results from dimension two to higher dimensions using elementary methods.

Contribution

It offers a new elementary proof for the density of a set related to affine curves in dimensions three and above, building on prior work by Cobeli, Zaharescu, and Foo.

Findings

The set is dense in [0,1]^n for n ≥ 3.

The proof simplifies previous complex arguments.

Extends known results from dimension 2 to higher dimensions.

Abstract

In Theorem 1 of Acta Arith. 99 (2001), 321-329, Cobeli and Zaharescu give a result about the distribution of the ${\bf F}_p$-points on an affine curve. An easy corollary to their theorem is that the set $\bigcup_p \lbrace (\frac{x_1}{p}, ...,\frac{x_n}{p}), 1 \leq x_i < p \text{and} \prod_{1 \leq i \leq n} x_i \equiv 1 \bmod{p} \rbrace$ is dense in $[ 0,1 ]^n$. In Integers 7 (2007), A7, Foo gives a elementary proof of that fact in dimension $2$. Following Foo's ideas, we give a similar proof in dimension greater than or equal to $3$.