Discrepancy in generalized arithmetic progressions

Abstract

Estimating the discrepancy of the set of all arithmetic progressions in the first N natural numbers was one of the famous open problems in combinatorial discrepancy theory for a long time, successfully solved by K. Roth (lower bound) and Beck (upper bound). They proved that D ( N ) = min χ max A | ∑ x ∈ A χ ( x ) | = Θ ( N 1 / 4 ) , where the minimum is taken over all colorings χ : [ N ] → { − 1 , 1 } and the maximum over all arithmetic progressions in [ N ] = { 0 , … , N − 1 } . Sumsets of k arithmetic progressions, A 1 + ⋯ + A k , are called k -arithmetic progressions and they are important objects in additive combinatorics. We define D k ( N ) as the discrepancy of the set { P ∩ [ N ] : P is a k -arithmetic progression } . The second author proved that D k ( N ) = Ω ( N k / ( 2 k + 2 ) ) and Přívětivý improved it to Ω ( N 1 / 2 ) for all k ≥ 3 . Since the probabilistic argument gives D k ( N ) = O ( ( N log N ) 1 / 2 ) for all fixed k , the case k = 2 remained the only case with a large gap between the known upper and lower bounds. We bridge this gap (up to a logarithmic factor) by proving that D k ( N ) = Ω ( N 1 / 2 ) for all k ≥ 2 . Indeed we prove the multicolor version of this result.