The exponent of discrepancy is at least 1. 0669 (Q1279909)

Let \(P\subset[0,1)^d\) be an \(n\)-point set and \(B_x=\prod^d_{k=1}[0,x_k)\subset[0,1)^d\). Let \[ D(x)=\text{vol}(B_x)-w(P\cap B_x), \] where \(w\) is a weighted function, for example \(w(x)=1/n\) \((n\in P)\). Let \(p^*\) be the infimum of \(p\) such that for all \(d\geq 1\) and \(\varepsilon>0\) there exists a weighted set of at most \(K(p)\varepsilon^{-p}\) points in \([0,1)^d\) with \(L_2\)-discrepancy at most \(\varepsilon\). The author proves that \(p^*\geq 1.0488 z\) by use of Roth's method on the lower estimation of discrepancy.