Abstract
We prove a query complexity variant of the weak polynomial Freiman–Ruzsa conjecture in the following form. For any ϵ>0, a set A⊂Zd with doubling K has a subset of size at least K−4ϵ|A| with coordinate query complexity at most ϵlog2|A|.We apply this structural result to give a simple proof of the “few products, many sums” phenomenon for integer sets. The resulting bounds are explicit and improve on the seminal result of Bourgain and Chang.
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