An elementary proof of a lower bound for the inverse of the star discrepancy

A central problem in discrepancy theory is the challenge of evenly distributing points $\{x_1,\dots ,x_n\}$ in $[0,1{]}^d$. Suppose a set is so regular that for some $\epsilon >0$ and all $y\in [0,1{]}^d$ the sub-region $[0,y]=[0,y_1]\times \cdots \times [0,y_d]$ contains a number of points nearly proportional to its volume and $$\mathrm{\forall }y\in [0,1{]}^d\mid \frac{1}{n}\mathrm{\#}\{1\le i\le n:x_i\in [0,y]\}-vol([0,y])\mid \le \epsilon ,$$ how large does $n$ have to be depending on $d$ and $\epsilon$? We give an elementary proof of the currently best known result, due to Hinrichs, showing that $n\gtrsim d\cdot {\epsilon }^{-1}$.