The analogue of Erdős-Turán conjecture in \(\mathbb Z_m\)
The particular conjecture of Erdős and Turán to which the title of the paper refers is that if \(A \subset \mathbb N\) is such that the support of \(1_A \ast 1_A\) contains every sufficiently large integer then it must be unbounded. Although the conjecture itself is far from resolution some interesting partial results are know. Of special interest is the result of Ruzsa that the modular analogue is false: writing \(R_m\) for the smallest number such that there is a set \(A \subset \mathbb Z/m\mathbb Z\) with \(A+A=\mathbb Z/m\mathbb Z\) and \(\| 1_A \ast 1_A\| _\infty \leq R_m\), Ruzsa showed that \(R_m = O(1)\). It is the purpose of the paper under review to show that \(R_m \leq 288\). The argument itself rests on two key facts. First, the author provides a careful analysis of the case when \(m=2p^2\) for some prime \(p\). In this instance he is able to give examples of sets \(A\) with \(A+A=\mathbb Z/m\mathbb Z\) and \(\| 1_A \ast 1_A\| _\infty \leq 48\). This is then combined with the fact that if \(m_1 < m_2 \leq 3m_1/2\) then \(R_{m_1} \leq 6R_{m_2}\) to complete the proof. The approach is combinatorial and the paper also includes a clutch of related open questions, although perhaps its most novel selling point is the video summary which may be found on youtube.
scientific article; zbMATH DE number 5368231
scientific article