Remark on height functions

Let $k$ be a number field and $V(k)$ an $n$-dimensional projective variety over $k$. We use the $K$-theory of a $C^{\ast }$-algebra $A_V$ associated to $V(k)$ to define a height of points of $V(k)$. The corresponding counting function is calculated and we show that it coincides with the known formulas for $n=1$. As an application, it is proved that the set $V(k)$ is finite, whenever the sum of the odd Betti numbers of $V(k)$ exceeds $n+1$. Our construction depends on the $n$-dimensional Minkowski question-mark function studied by Panti and others.