Bivariate fluctuations for the number of arithmetic progressions in random sets

Abstract

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 ∈ [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 → +∞.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) → +∞. 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 nontrivial is characterised by the asymptotic behaviour (as n → +∞) of the threshold function ψ ℓ = ψ 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.

Normal limiting distributions for systems of linear equations in random sets

“…We write if n(1 − p) → ∞ and np c(A) → ∞. Andréani, Koch and Liu [2]. In fact, they prove not only results on the limiting distribution of the number of k-term arithmetic progressions in [n] p even when k is unbounded (but sublogarithmic), they also establish a bivariate central limit theorem for the joint distribution when considering the counts of two distinct progression lengths.…”

Deviation probabilities for arithmetic progressions and irregular discrete structures

“…Let us mention that our results also give central limit theorems for D H (B m ) and D H (B p ). We remark that the central limit theorem (and even a bivariate version) is already known for the special case of arithmetic progressions [3]. However, we are not aware of a central limit theorem in the more general context of random induced subhypergraphs.…”

“…See [19] and [22] for more detailed discussions of lower tail problems. The grey lines on the left represent deviations of the order of the standard deviation and so are covered by the central limit theorem [3]. The blue areas ought to belong to the "Normal" regime in which we would expect that r(N, p, δ) to be of the form (1 + o(1))3δ 2 N pN/56(1 − p).…”

On a class of random sets of positive integers

“…The random counterpart of Erdős and Turán's problems attracts many attentions(cf. [4,2,7,1]). However, it seems that our random set E X is different from these mentioned ones, and it is natural to consider arithmetic progressions properties of E X .…”