We contemplate $x_1,..,x_N$ factors within the aircraft $mathbb{R}^2.$

We outline the sum

$$F(x):=frac{1}{N^2}sum_{i=1}^N sum_{j neq i} vert x_i-x_j vert^{-2}.$$

I’m searching for an announcement of the next kind:

If $F(x) le 1$ then $vert x vert^2 gtrapprox N$ the place $gtrapprox$ implies that in some sense such a degree configuration ought to have norm bigger than $sqrt{N}.$

Here’s a heuristic argument why one thing like this could true.

If we place factors $x_1,..,x_N$ uniformly on the circle or radius $sqrt{N}$ then arguably the factors $x_1,…,x_N$ are fairly removed from one another.

In that case nonetheless

$$F(x):=frac{1}{N^3}sum_{i=1}^N sum_{j neq i} vert y_i-y_j vert^{-2}$$ the place $y_i$ are actually on the unit circle.

Since they’re uniformly distributed, now we have that

$$F(x):=frac{1}{N^2} sum_{j neq 1} vert y_1-y_j vert^{-2}.$$

Nevertheless, it’s well-known that this final sum $sum_{j neq 1} vert y_1-y_j vert^{-2}$ is understood to be of order $N^2$, evaluate with this one and comparable questions. So on this vital case, we exactly one thing of order $1$.

So can we get a very good description of factors the place $F(x) le 1?$