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On the sum of dilations of a set

Volume 164 / 2014

Antal Balog, George Shakan Acta Arithmetica 164 (2014), 153-162 MSC: 11B13, 05B10, 11B30. DOI: 10.4064/aa164-2-5


We show that for any relatively prime integers $1\leq p< q$ and for any finite $A \subset \mathbb {Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$


  • Antal BalogAlfréd Rényi Institute of Mathematics
    P.O. Box 127
    1364 Budapest, Hungary
  • George ShakanDepartment of Mathematics
    University of Wyoming
    Laramie, WY 82072, U.S.A.

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