Ah but the conjecture isn't about what happens “on average” or for a “random set” of integers. It's about what is forced to happen in every possible case. (In the “random” case there are probably very few or no duplicates anyway, as both a+b=c+d and ab=cd will be unlikely for completely random integers — note that they say d is exactly a+b-c or ab/c respectively — the conjecture is about limiting what you could possibly arrange to happen, non-randomly.)
Basically, when you take integers like {1, 2, 3, 4} or {1, 3, 5, 7} (so that there are additive relations between them) (e.g. small integers), there tends to be a lot of duplicates in the addition table. But if you take integers like {1, 2, 4, 8} or {1, 3, 9, 27} (so that there are multiplicative relations between them) (e.g. very similar prime factorizations), there tend to be a lot of duplicates in the multiplication table. In other words, it's very easy to create sets with either lots of duplicates in the addition table or lots of duplicates in the multiplication table.
The conjecture is not about which kind of set is more common (so it doesn't matter even if it's true that there are a lot “more” arithmetic progressions than geometric ones, in whatever sense), but rather says there is no way you can create a set with both sorts of collisions — it's saying that if a set has many additive coincidences it can't have many multiplicative coincidences and vice-versa.
Basically, when you take integers like {1, 2, 3, 4} or {1, 3, 5, 7} (so that there are additive relations between them) (e.g. small integers), there tends to be a lot of duplicates in the addition table. But if you take integers like {1, 2, 4, 8} or {1, 3, 9, 27} (so that there are multiplicative relations between them) (e.g. very similar prime factorizations), there tend to be a lot of duplicates in the multiplication table. In other words, it's very easy to create sets with either lots of duplicates in the addition table or lots of duplicates in the multiplication table.
The conjecture is not about which kind of set is more common (so it doesn't matter even if it's true that there are a lot “more” arithmetic progressions than geometric ones, in whatever sense), but rather says there is no way you can create a set with both sorts of collisions — it's saying that if a set has many additive coincidences it can't have many multiplicative coincidences and vice-versa.