...umptions of Theorem 1 and 2, since by the Erdős–Rényi “pure heads” theorem with probability one such a sequence will have gaps for mk+1 −mk of order roughly log k, infinitely often (see for example =-=[17, 26]-=-). Thus we define a second sequence (nk)k≥1 = (nk(ω))k≥1 which for a given ω and corresponding random sequence (mk)k≥1 contains all the number 2k − 1, k ≥ 1, and 2mk, k ≥ 1, sorted in increasing order...