A note on the length of maximal arithmetic progressions in random subsets

Let U (n) denote the maximal length arithmetic progression in a non-uniform random subset of {0, 1} n , where 1 appears with probability pn. By using dependency graph and Stein-Chen method, we show that U (n) − cn ln n converges in law to an extreme type distribution with ln pn = −2/cn. Similar result holds for W (n) , the maximal length aperiodic arithmetic progression (mod n). An arithmetic progression is a sequence of numbers such that the difference of any two successive members of the sequence is a constant. A celebrated result of Szemerédi [5] says that any subset of integers of positive upper density contains arbitrarily long arithmetic progressions. The recent work [6] reviews some extremal problems closely related with arithmetic progressions and prime sequences, under the name of the Erdös-Turán conjectures, which are known to be notoriously difficult to solve. Let ξ 1 , ξ 2 , · · · , ξ n be a uniformly chosen random word in {0, 1} n and Ξ n be the random set consisting elements i such that ξ i = 1. Benjamini et al. [3] studies the length of maximal arithmetic progressions in Ξ n. Denote by U (n) the maximal length arithmetic progression in Ξ n and W (n) the maximal length aperiodic arithmetic progression (mod n) in Ξ n. They show, among others, that the expectation of U (n) and W (n) is roughly 2 ln n/ ln 2. In view of the random graph theory [4], a natural extension of [3] is to consider non-uniform random subset of {0, 1} n , which is the main interest of this note. Let ξ i = 1 with probability p n and ξ i = 0 with probability 1 − p n , where p n ∈ [0, 1] is a function of n. Following [3], the key to our work is to construct proper dependency graph and apply the Stein-Chen method of Poisson approximation (see e.g. [1,4]). Our result implies that, in the non-uniform scenarios, the expectation of U (n) and W (n) is roughly c n ln n, with ln p n = −2/c n. Obviously, taking p n ≡ 1/2 and c n ≡ 2/ ln 2, we then recover the main result of Benjamini et al. The rest of the note is organized as follows. We present the main results in Section …