Frantzikinakis' random-difference conjecture for Szemerédi's theorem

Let $S\subset\mathbb{N}$ be chosen at random with $\mathbb{P}(d\in S)=\omega(1/d)$. A set of positive upper density means a set $A\subset\mathbb{N}$ for which $\limsup_{N\to\infty}|A\cap[N]|/N>0$; a $k$-term arithmetic progression is a sequence $x,x+d,\ldots,x+(k-1)d$ with $d\in S$. Frantzikinakis' conjecture. Almost surely, every subset of $\mathbb{N}$ with positive upper density contains a $k$-term arithmetic progression whose common difference lies in $S$. This predicts that a random set of differences with probabilities just larger than order $1/d$ suffices for Szemerédi's theorem, extending known results for denser random difference sets; the statement is presented as a conjectural reformulation of work of Frantzikinakis and others, and its resolution is not established here.