Weighted exponential sums and discrepancy (Q908956)

Let \(P=(p_ n)_{n\to \infty}\) with \(p_ n\geq 0\), \(p_ 1>0\) and \(S_ N=\sum^{N}_{n=1}p_ n \to \infty\) as \(N\to \infty\). For real numbers \(0\leq x_ 1,...,x_ N<1\) let \[ D^*_ N(P)=D^*_ N(P;x_ 1,...,x_ N)=\sup_{0<t\leq 1}| (1/S_ N)\sum_{(n=1,...,N;\quad x_ n<t)}p_ n-t| \] be their P-discrepancy. If \(C_ N(P)\) denotes the least constant such that \[ | \sum^{N}_{n=1}p_ ne^{2\pi ix_ n}| \quad \leq \quad C_ N(P) D^*_ N(P) S_ N \] for all such sequences \(x_ 1,...,x_ N\), then the inequalities \[ 4-(12\pi m_ N/S_ N)^{2/3}\leq C_ N(P)\leq 4-(\sqrt{3}m_ N/2S_ N)^ 2 \] will be proved. Here \(m_ N\) denotes the maximum of \(p_ 1,...,p_ N\). The exponents are in general best possible. Furthermore the estimation \(C_ N(P)\leq 4-1.15/N^{2/3}\) will be shown.