Discrepancy of Sums of two Arithmetic Progressions

Estimating the discrepancy of the hypergraph of all arithmetic progressions in the set [N ] = {1, 2, . . . , N} was one of the famous open problems in combinatorial discrepancy theory for a long time. An extension of this classical hypergraph is the hypergraph of sums of k (k ≥ 1 fixed) arithmetic progressions. The hyperedges of this hypergraph are of the form A1 + A2 + . . . + Ak in [N ], where the Ai are arithmetic progressions. For this hypergraph Hebbinghaus (2004) proved a lower bound of Ω(Nk/(2k+2)). Note that the probabilistic method gives an upper bound of order O((N logN)1/2) for all fixed k. Př́ıvětivý improved the lower bound for all k ≥ 3 to Ω(N1/2) in 2005. Thus, the case k = 2 (hypergraph of sums of two arithmetic progressions) remained the only case with a large gap between the known upper and lower bound. We bridge this gap (up to a logarithmic factor) by proving a lower bound of order Ω(N1/2) for the discrepancy of the hypergraph of sums of two arithmetic progressions.