Discrepancy of sums of three arithmetic progressions (Q813920)

Let \((X,{\mathcal F})\) be a set system on a finite set. The author proves that for the (worst-case) discrepancy \(\text{disc}({\mathcal F})=\min_{\chi:X\to\{-1,1\}}\max_{S\in{\mathcal F}}\left| \sum_{x\in S}\chi(x)\right| \) of the set system \({\mathcal F}\) formed by all sums of three arithmetic progressions on \(X=\{0,1,2,\dots,n\}\) we have \(\text{disc}({\mathcal F})=\Omega(n^{1/2})\).