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

Abstract

In Theorem 1 of Acta Arith. 99 (2001), 321-329, Cobeli and Zaharescu give a result about the distribution of the ${\mathbf{F}}_p$-points on an affine curve. An easy corollary to their theorem is that the set ${\bigcup }_p\{(\frac{x_1}{p},...,\frac{x_n}{p}),1\le x_i<p\text{and}{\prod }_{1\le i\le n}{x}_i\equiv 1\mathrm{mod}p\}$ 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 $3$.