Antal Balog, George Shakan
Published 2013-11-03, updated 2013-11-19Version 2
We show that for any relatively prime integers $1\leq p<q$ and for any finite $A \subset \mathbb{Z}$ one has $$|p \cdot A + q \cdot A | \geq (p + q) |A| - (pq)^{(p+q-3)(p+q) + 1}.$$
Strictly chained (p,q)-ary partitions
Note on a theorem of Birch and Erdős
Number of solutions to $a^x + b^y = c^z$, A Shorter Version
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