Query complexity and the polynomial Freiman-Ruzsa conjecture
Abstract
We prove a query complexity variant of the weak polynomial Freiman-Ruzsa conjecture in the following form. For any $\epsilon > 0$, a set $A \subset \mathbb{Z}^d$ with doubling $K$ has a subset of size at least $K^{-\frac{4}{\epsilon}}|A|$ with coordinate query complexity at most $\epsilon \log_2 |A|$. 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.
- Publication:
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arXiv e-prints
- Pub Date:
- March 2020
- DOI:
- 10.48550/arXiv.2003.04648
- arXiv:
- arXiv:2003.04648
- Bibcode:
- 2020arXiv200304648Z
- Keywords:
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- Mathematics - Number Theory;
- 11B30
- E-Print:
- Restructured the paper for a more coherent exposition and extended the proofs