# Simultaneous Linear Discrepancy for Unions of Intervals

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## Abstract

Lovász proved (see [7]) that given real numbers *p*_{1},..., *p*_{n}, one can round them up or down to integers ϵ_{1},..., 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 \(1-\frac{1}{n+1}\). 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 \(\left(1-\frac{d}{n+1}\right)d\). 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## Preview

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## References

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© János Bolyai Mathematical Society and Springer-Verlag Berlin Heidelberg 2018