Number of arithmetic progressions in dense random subsets of $\mathbb{Z}/n\mathbb{Z}$

Abstract

We examine the behavior of the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/n\mathbb{Z}$. We prove that if a set is chosen by including each element of $\mathbb{Z}/n\mathbb{Z}$ independently with constant probability $p$, then the resulting distribution of $k$-term arithmetic progressions in that set, while obeying a central limit theorem, does not obey a local central limit theorem. The methods involve decomposing the random variable into homogeneous degree $d$ polynomials with respect to the Walsh/Fourier basis. Proving a suitable multivariate central limit theorem for each component of the expansion gives the desired result.