Arithmetic progressions in sumsets of random sets - Miyazaki, Rafael Kazuhiro (2023)
Given a set A, its sumset A+A is defined as the set of all sums of two elements, not necessarily distinct, in A. Given a function p \\colon \\N \\to [0,1], we consider the sequence of independent random sets \\{A_n\\}_{n\\in \\N}, where A_n is obtained by choosing independently each integer 1\\le i \\le n with probability p(n). We employ the classical probabilistic tools of the first and second moment methods as well as a recently proven theorem of Park and Pham, formerly known as the Kahn--Kalai Conjecture, regarding the relationship between the threshold function and the expectation threshold of increasing properties in order to find lower and upper bounds for the threshold for the existence of arithmetic progressions of m(n) elements in the sumset of the random set A_n.