# co.combinatorics – On a uniformity of a set of points

We write $$D(r) subset mathbb{R}^2$$ for the unit disk centered at the origin with radius $$r>0$$.
Let $$n$$ be an positive integer. We take $$n$$ different points $${p_k}_{k=1}^n$$ on $$D(1)$$, and set $$V_n=bigsqcup_{k=1}^n{p_k}_{k=1}^n$$

We now consider the following condition: there exists $$Cge 1$$ such that for any $$r in (0,1)$$ and any $$a in mathbb{R}^2$$ with $$|e|=1$$,
begin{align*} (ast)quad sum_{x in V_n cap D(r)}|x|^2 & le C sum_{x in V_n cap D(r)}langle x,arangle^2. end{align*}
Here, $$|cdot|$$ denotes the Euclidean norm on $$mathbb{R}^2$$ and $$langle cdot,cdotrangle$$ the standard inner product on $$mathbb{R}^2$$.

I think $$(ast)$$ means that $${p_k}_{k=1}^n$$ are uniformly placed (in a sense). For example, can we obtain the positive constant $$C$$ such that $$#(V_n cap D(r/2)) ge C times #(V_n cap D(r))$$ for any $$r in (0,1) ?$$