On a generalisation of Roth's theorem for arithmetic progressions and applications to sum-free subsets (Q2845628)

Roth's theorem on arithmetic progressions states that for any \(0 < \alpha < 1\), if \(N \geq \exp(\exp(C\alpha^{-1}))\) for some constant \(C\), then any subset \(A \subset [1, N]\) with \(|A| \geq {\alpha}N\) contains a non-trivial arithmetic progression of length \(3\). We say a set of the form \(\{n_{i} + n_{j} + a: 1 \leq i \leq j \leq d\}\), where \(a, n_{1}, \dots ,n_{d}\) are natural numbers form a \(d\)-configuration. A \(d\)-configuration is non-trivial if for all \(i \neq j\), \(n_{i} \neq n_{j}\). In this paper the author generalize Roth's theorem to \(d\)-configurations. In particular she proves that for any \(0 < \alpha < 1\) if \(N \geq \exp(\exp((\frac{C'}{\alpha})^{(d(d+1)-1)})))\) for some constant \(C'\), then any subset \(A \subset [1, N]\) with \(|A| \geq {\alpha}N\) contains a non-trivial \(d\)-configuration. In the proof she uses the Gowers uniformity norms and the density increment argument. Let \(A\) and \(B\) two finite sets of real numbers. We say \(B\) is sum-free with respect to \(A\) if the set \(\{b+b': b,b' \in B, b \neq b'\} \cap A = \emptyset\). Applying the above result, the author improves on a theorem about sum-free subsets of Sudakov, Szemerédi and Vu [\textit{B. Sudakov} et al., Duke Math. J. 129, No. 1, 129--155 (2005; Zbl 1155.11346)] by proving that any set of \(n\) integers contains a sum-free subset of size at least \(\log n(\log \log \log n)^{1/32772-o(1)}\).