Finding a Low-dimensional Piece of a Set of Integers
Finding a Low-dimensional Piece of a Set of Integers
We show that a finite set of integers |$A \subseteq \mathbb{Z}$| with |$|A+A| \le K |A|$| contains a large piece |$X \subseteq A$| with Freĭman dimension |$O(\log K)$|, where large means |$|A|/|X| \ll \exp(O(\log^2 K))$|. This can be thought of as a major quantitative improvement on Freĭman’s dimension lemma; or …