Let $(X_n)_{n\in\mathbb{N}}$ be a sequence of independent random variables with $P(X_n=1)=\frac{1}{logn}$ and $P(X_n=0)=1-P(X_n=1), for $ $n\geq2$.
Show that almost surely the set $ S(ω)=\{{n\in\mathbb{N }:X_n(ω)=1}\} $ is a set of recurrence.
What I have shown is that a subset of $\mathbb{N}$ is a set of recurrence if it contains arbitrarily large arithmetic progressions starting from 0, i.e. for every k $\in\mathbb{N}$ there exists some n $\in\mathbb{N}$ such that the arithmetic progression {n,2n,...,kn} belongs to the subset.
So, all I have to prove is that S(ω) almost surely contains arbitrarily large arithmetic progressions starting from zero, that is $P(\bigcap_{k\in\mathbb{N}}\bigcup_{n \in\mathbb{N}}${$X_n(ω)=1,X_{2n}(ω)=1,...,X_{kn}(ω)=1\})=1$.
Now, this holds if for every $k\in\mathbb{N}$ we have that $$P(\bigcup_{n \in\mathbb{N}}\{X_n(ω)=1,X_{2n}(ω)=1,...,X_{kn}(ω)=1\})=1$$ The independence implies that P({$X_n(ω)=1,X_{2n}(ω)=1,...,X_{kn}(ω)=1$}$)=\prod_{i=1}^k\frac{1}{log(in)}$
We can also consider the sum $\sum_{n=1}^\infty X_n(ω)X_{2n}...X_{kn}(ω) $ which is the number of k-term arithmetic progressions contained in S(ω). Then we have to show that $\sum_{n=1}^\infty X_n(ω)X_{2n}(ω)...X_{kn}(ω) >0$ almost surely.
What we know is that the expected value $$\mathbf{E}[\sum_{n=1}^\infty X_n(ω)X_{2n}(ω)•••X_{kn}(ω)]=\sum_{n=1}^\infty \mathbf{E}[X_n(ω)X_{2n}(ω)•••X_{kn}(ω)]=\sum_{n=1}^\infty \prod_{i=1}^k\frac{1}{log(in)}=\infty$$
Any help would be much appreciated!