Maximal Arithmetic Progressions in Random Subsets

TL;DR

This paper investigates the distribution and almost sure behavior of the maximal length of arithmetic progressions in random subsets of {0,1}^N, revealing convergence properties and distributional limits using advanced probabilistic methods.

Contribution

It establishes the limiting distribution of the maximal arithmetic progression length in random subsets and compares almost sure convergence for different definitions.

Findings

U(N) - 2 log(N)/log(2) converges in law to an extreme distribution

U(N)/log(N) converges almost surely to 2/log(2)

W(N)/log(N) does not converge almost surely, with limsup at least 3/log(2)

Abstract

Let U(N) denote the maximal length of arithmetic progressions in a random uniform subset of {0,1}^N. By an application of the Chen-Stein method, we show that U(N)- 2 log(N)/log(2) converges in law to an extreme type (asymmetric) distribution. The same result holds for the maximal length W(N) of arithmetic progressions (mod N). When considered in the natural way on a common probability space, we observe that U(N)/log(N) converges almost surely to 2/log(2), while W(N)/log(N) does not converge almost surely (and in particular, limsup W(N)/log(N) is at least 3/log(2)).