Some properties of lower level-sets of convolutions (Q2908126)

For a finite abelian group \(G\), written additively, there are various conditions on a subset \(A\) that guarantee that \(A+A\) is nearly all of \(G\). The aim of the paper is to investigate conditions on \(A+A\) itself that might guarantee that \(A+A\) is (nearly) all of \(G\). Specifically, the considered conditions are along the lines that \(A+A\) contains a certain (small) set \(S\).NEWLINENEWLINEIn particular, the following result is obtained. Let \(G\) be a finite abelian group of order \(N\), and let \(0 < \theta \leq 1\) and \(\delta, \varepsilon > 0\). Then, there exists a set \(S \subset G\) with NEWLINE\[NEWLINE |S| \ll \varepsilon^{-2} \delta^{-6} \theta^{-10} (\log N - \log ( \delta \theta \varepsilon)), NEWLINE\]NEWLINE such that if \(A \subset G \), \(|A| = \theta_0N \geq \theta N \) satisfies \(1_A \ast 1_A (x) > \delta \theta_0^2 N\) for every \(x \in S\), then \(|A + A |> (1- \varepsilon) N\). (The convolution is the unnormalized convolution.)NEWLINENEWLINEFurthermore, it is then shown that any set whose non-trivial Fourier coefficients are small (in a suitable and precise sense) works for this purpose.NEWLINENEWLINEIn addition, results on the existence of long arithmetic progressions and other structures (Bohr neighborhoods) in level sets of \(1_A \ast 1_A (x)\) are obtained; a level set is a set of the form \(\{ x \in G : 1_A \ast 1_A (x) \leq \gamma |G| \}\) for some parameter \(\gamma\).NEWLINENEWLINEFinally, several conjectures related to this circle of ideas are presented.