Maximal arithmetic progressions in random subsets

Let U^(N) denote the maximal length of arithmetic progressions in a random uniform subset of (0, 1)N. By an application of the Chen-Stein method, we show that U^(N) − 2 logN/ log² converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W^(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U^(N)/ logN converges almost surely to 2/log 2, while W^(N)/ logN does not converge almost surely (and in particular, limsup W^(N)/ logN ≥ 3/ log 2).