Bivariate fluctuations for the number of arithmetic progressions in random sets

We study arithmetic progressions {a, a + b, a + 2b, ..., a + (l- 1)b}, with l >= 3, in random subsets of the initial segment of natural numbers [n] := {1, 2, ..., n}. Given p is an element of [0, 1] we denote by [n](p) the random subset of [n] which includes every number with probability p, independently of one another. The focus lies on sparse random subsets, i.e. when p = p(n) = o(1) as n -> +infinity. Let X-l denote the number of distinct arithmetic progressions of length l which are contained in [n](p). We determine the limiting distribution for X-l not only for fixed l >= 3 but also when l = l(n) -> +infinity with l = o(log n). The main result concerns the joint distribution of the pair (X-l, X-l',), l > l', for which we prove a bivariate central limit theorem for a wide range of p. Interestingly, the question of whether the limiting distribution is trivial, degenerate, or non-trivial is characterised by the asymptotic behaviour (as n -> +infinity) of the threshold function psi(l) = psi(l)(n) := np(l-1) l. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.