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When  A  and  B  are subsets of the integers in [1,  X ] and [1,  Y ], respectively, with | A | ⩾  α X  and | B | ⩾  β Y , we show that the number of rational numbers expressible as  a / b  with ( a ,  b ) in  A  ×  B  is ≫ ( α β ) 1 + ϵ  XY  for any ϵ > 0, where the implied constant depends on ϵ alone. We then construct examples that show that this bound cannot, in general, be improved to ≫  α β XY . We also resolve the natural generalization of our problem to arbitrary subsets  C  of the integer points in [1,  X ] × [1,  Y ]. Finally, we apply our results to answer a question of Sárközy concerning the differences of consecutive terms of the product sequence of a given integer sequence.

The number of rational numbers determined by large sets of integers