Number of arithmetic progressions in dense random subsets of ℤ/nℤ
- Resource Type
- Authors
- Mehtaab Sawhney; Ashwin Sah; Ross Berkowitz
- Source
- Israel Journal of Mathematics. 244:589-620
- Subject
- Set (abstract data type)
Degree (graph theory)
Distribution (number theory)
General Mathematics
Basis function
Arithmetic
Element (category theory)
Constant (mathematics)
Random variable
Central limit theorem
Mathematics
- Language
- ISSN
- 1565-8511
0021-2172
We examine the behavior of the number of k-term arithmetic progressions in a random subset of ℤ/nℤ. We prove that if a set is chosen by including each element of ℤ/nℤ 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.