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If $A$ is a subset of an abelian group with $|A+A|\leq K|A|$ then one can show that $A$ contains a large cube of size depending on $K$. Here a cube is a set of the form $$C=\{a_0+\sum_{i=1}^d e_ia_i:e_i\in\{0,1\}\}$$.

Usually, if we take sumsets or difference sets, finding large arithmetic structures becomes easier. My question is, given a bound on $|A+A|$, what is the largest cube one can find in, say, $10A-10A$?

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