Discrepancy in generalized arithmetic progressions

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(X) max(A) vertical bar Sigma(x is an element of A) chi (x)vertical bar = Theta(N-1/4). where the minimum is taken over all colorings chi : [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 boolean AND [N] : P is a k-arithmetic progression}. The second author proved that D-k(N) = Omega(Nk/(2k+2)) and Privetivy improved it to Omega(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) = Omega(N-1/2) for all k >= 2. Indeed we prove the multicolor version of this result. (C) 2009 Elsevier Ltd. All rights reserved.