On the sum of dilations of a set Antal Balog; George Shakan — 2014 Acta Arithmetica We show that for any relatively prime integers 1 ≤ p < q and for any finite A ⊂ ℤ one has | p · A + q · A | ≥ ( p + q ) | A | - ( p q ) ( p + q - 3 ) ( p + q ) + 1 .