Discrepancy of line segments for general lattice checkerboards (Q519959)

In this paper new lower bounds for functionals related to the so-called ``checker-board discrepancy'' are obtained. Let \(T\subseteq\mathbb{R}^2\) be a lattice (of area \(1\)), \(G\subseteq T\) a finite subset of lattice points and the ``coloring function'' \(f:\mathbb{R}^2\to\mathbb{C}\) be given by \[ f(x)=\sum_{g\in G} z_g\chi_Q(x-g), \] where \(Q\) is a measureable fundamental domain of \(T\) and \(z_g\in\mathbb{C}\) are arbitrary weights; \(\chi_Q\) denoting the indicator function of \(Q\). Then there is a straight line \(S\) such that the line integral of \(f\) along \(S\) can be estimated from below as follows: \[ \Bigl| \int_S f\Bigr| \gg (\text{diam}\,G)^{-1/2} \Bigl(\sum_{g\in G} | z_g| ^2\Bigr)^{1/2}. \]