We show that for any integer n ≥ 1 and real ε > 0, the union of n arithmetic progressions with pairwise distinct differences, each of length n, contains at least c (ε)n2-ε elements, where c (ε) is a positive constant depending only on ε. This estimate is sharp in the sense that the assertion becomes invalid for ε = 0. We also obtain estimates for the "asymmetric case" where the number of progressions is distinct from their lengths.
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© 2014 Society for Industrial and Applied Mathematics.