Number of arithmetic progressions in dense random subsets of DOUBLE-STRUCK CAPITAL Z/nDOUBLE-STRUCK CAPITAL Z

We examine the behavior of the number of k-term arithmetic progressions in a random subset of DOUBLE-STRUCK CAPITAL Z/nDOUBLE-STRUCK CAPITAL Z. We prove that if a set is chosen by including each element of DOUBLE-STRUCK CAPITAL Z/nDOUBLE-STRUCK CAPITAL 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.