Abstract
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,ωN) 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 ωN.
We give lower and upper bounds for βN (f) in three special cases, together with some results of a more general type.
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References
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© 1989 Springer-Verlag Berlin Heidelberg
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Wagner, G. (1989). On an Imbalance Problem in the Theory of Point Distribution. In: Halász, G., Sós, V.T. (eds) Irregularities of Partitions. Algorithms and Combinatorics 8, vol 8. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61324-1_14
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DOI: https://doi.org/10.1007/978-3-642-61324-1_14
Publisher Name: Springer, Berlin, Heidelberg
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