Simultaneous Linear Discrepancy for Unions of Intervals

Abstract

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 $1-\frac{1}{n+1}$. 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.