Simultaneous Linear Discrepancy for Unions of Intervals

Lovász proved (see [7]) that given real numbers p1,..., pn, one can round them up or down to integers ϵ1,..., ϵn, in such a way that the total rounding error over every interval (i.e., sum of consecutive pi’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.