On the postage stamp problem with three stamp denominations, III.
Let K be a finite set of lattice points in a plane. We prove that if |K| is sufficiently large and |K+K| < (4 - 2/s)|K| - (2s-1), then there exist s - 1 parallel lines which cover K. We also obtain some more precise structure theorems for the cases s = 3 and s = 4.
We show that for any relatively prime integers 1 ≤ p < q and for any finite A ⊂ ℤ one has .