On an Imbalance Problem in the Theory of Point Distribution

We consider the following problem: Let f be a 2π-periodic integrable function satisfying $$\int_{0}^{2\pi} f(x)dx=0$$. Given an N-tuple of points $${\omega _N} = \left\{ {{x_1},{x_2},...,{x_N}} \right\}$$ on [0, 2π), denote by Pos(f,ω_N) the set of all x∈[0,2π) for which $$\sum\ _{j=1}^{N}f(x-x_j)\geq 0$$ is true. Let $$\beta _N(f)=inf\ m(Pos(f,\omega _N))$$ where m denotes Lebesgue measure on [0,2π) and the infimum is taken over all N-tuples ω_N.We give lower and upper bounds for β_N (f) in three special cases, together with some results of a more general type.