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 $r in (0,1) ?$