Products of two elements of Granville's random model in residue classes

Source abstract

Erdős, Odlyzko and Sárközy conjectured that every reduced residue class modulo $q$ contains a product of two primes not exceeding $q$; their conjecture remains open even under the GRH. We prove that its analogue holds, unconditionally and in the wider range $q^Θ$ for any fixed $Θ> 3/4$, in Granville's random model of the primes. Almost surely, every reduced class modulo every sufficiently large modulus is representable, and the number of representations is asymptotic to its expected order. The choice of the model is decisive; the corresponding statement in the random prime model of Banks, Ford and Tao fails with positive probability, an elementary local defect that Granville's discard step removes.