On additive bases of sets with small product set

We prove that finite sets of real numbers satisfying $|AA| \leq |A|^{1+ε}$ with sufficiently small $ε> 0$ cannot have small additive bases nor can they be written as a set of sums $B+C$ with $|B|, |C| \geq 2$. The result can be seen as a real analog of the conjecture of Sárközy that multiplicative subgroups of finite fields of prime order are additively irreducible.

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