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 log N/log2 converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W-(N) of arithmetic prorpgressions (mod N). When considered in the natural way on a common probability space, we observe that U-(N)/logN converges almost surely to 2/log2, while W-(N)/logN does not converge almost surely (and in particular, lim sup W-(N)/log N >= 3/log 2).