# Equidistribution relative to the Zariski topology

I'm interested in measuring the distribution of an ordered set of points relative to the Zariski topology. Possibly this is a standard idea (with different terminology?) in algebraic geometry. Any pointers to the literature would be appreciated. Here's what I have in mind.

Let $X\subset\textbf{P}^N$ be a projective variety defined over an algebraically closed field $k$. For any finite set of points of $X$, define the $X$-degree of the set to be the minimal degree of a homogeneous polynomial $F\in k[X_0,\ldots,X_n]$ that does not vanish identically on $X$, but vanishes at all of the points in the set. (In other words, the minimal degree of a hypersurface that goes through the points, but does not contain $X$.)

Now take a sequence of points $S=(P_1,P_2,P_3,\ldots)\subset X$, and for each $n\ge1$, let $S[n]=\{P_1,P_2,\ldots,P_n\}$. Intuitively, if $\deg_X S[n]$ grows quickly, then the set $S$ is well distributed relative to the Zariski topology. A rough guess is that one might define equidistribution by the condition $$\lim_{n\to\infty} \frac{\log\deg_XS[n]}{\log n} = \frac{1}{\dim X}.$$ (It is not hard to check that the limsup of the left-hand side is at most $\frac{1}{\dim X}$.)

[Edit: value of limit fixed as suggested by JSE]