On a problem of Erdős in the theory of irregularities of distribution (Q1076718)

Let \(S\subset\mathbb{R}^2\) be a completely arbitrary infinite discrete set on the plane \(\mathbb{R}^2\). For each circular disc \(D\) of radius \(r\) let \(Z(D)\) denote the number of points of \(S\) which lie in \(D\). Write \(f(D) = | Z(D)-\pi \cdot r^2|\) and \(f(r)=\max f(D)\) where the maximum is taken over all discs \(D\) of radius \(r\). In 1964 Erdős raised the question of proving the relation \(f(r)\to \infty\) as \(r\to \infty\). The purpose of this paper is to prove this conjecture. The proof is based on a Fourier transform approach.