Simultaneous Linear Discrepancy for Unions of Intervals



Lovász proved (see [7]) that given real numbers p1,..., p n , one can round them up or down to integers ϵ1,..., ϵ n , in such a way that the total rounding error over every interval (i.e., sum of consecutive p i ’s) is at most 1-1/n+1. Here we show that the rounding can be done so that for all \(d = 1,...,\left\lfloor {\frac{{n + 1}}{2}} \right\rfloor \), the total rounding error over every union of d intervals is at most (1- d/n+1) d. This answers a question of Bohman and Holzman [1], who showed that such rounding is possible for each value of d separately.

Mathematics Subject Classification (2000)

05C65 11K38 


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Copyright information

© János Bolyai Mathematical Society and Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of MathematicsTechnion-Israel Institute of TechnologyHaifaIsrael

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