Bivariate fluctuations for the number of arithmetic progressions in random sets

We study arithmetic progressions $\{a,a+b,a+2b,\dots,a+(\ell-1) b\}$, with $\ell\ge 3$, in random subsets of the initial segment of natural numbers $[n]:=\{1,2,\dots, n\}$. Given $p\in[0,1]$ 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 $n\to+\infty$. Let $X_\ell$ denote the number of distinct arithmetic progressions of length $\ell$ which are contained in $[n]_p$. We determine the limiting distribution for $X_\ell$ not only for fixed $\ell\ge 3$ but also when $\ell=\ell(n)\to+\infty$. The main result concerns the joint distribution of the pair $(X_{\ell},X_{\ell'})$, $\ell>\ell'$, 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 $n\to+\infty$) of the threshold function $\psi_\ell=\psi_\ell(n):=np^{\ell-1}\ell$. The proofs are based on the method of moments and combinatorial arguments, such as an algorithmic enumeration of collections of arithmetic progressions.