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 A 1 + A 2 + ⋯ + A k in [ N ] , where the A i are arithmetic progressions. The case k = 2 (hypergraph of sums of two arithmetic progressions) is 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 ( N 1 / 2 ) for the discrepancy of the hypergraph of sums of two arithmetic progressions.