Davenport's theorem in the theory of irregularities of point distribution

We study distributions ${\mathscr D}_N$ of $N$ points in the unit square $U^2$ with a minimal order of the $L_2$-discrepancy ${\mathscr L}_2[{\mathscr D}_N]<C(\log N)^{1/2}$, where the constant $C$ is independent of $N$. We introduce an approach using Walsh functions that admits generalization to higher dimensions