Arithmetic progressions in sets of small doubling (Q2810742)

Let \(A\) be a finite set in a commutative group. The paper presents several results of the following kind: if \( |A+A| < K |A| \), then \(A\) and \(|A+A|\) contain arithmetic progressions, the length depending on the size of \(K\). Remarkably, the results are almost as strong as known under the stronger assumption that the density of \(A\) is \(1/K\). The existence of a 3-term progression is shown for NEWLINE\[NEWLINE K < c ( \log |A| ) ( \log\log |A| )^{-7} , NEWLINE\]NEWLINE and \(A+A\) is shown to contain a progression or a coset of size \( \exp c \left( (\log |A|)/ (K ( \log K)^3 \right)^{1/2} \).