Sumsets being squares
An old problem of Euler on sumsets \(A + B\) asks for which \(A\) and \(B\) all elements of \(A + B\) are integer squares. Note that if \(| A| = 2\), then the size of \(| B| \) is bounded by a divisor function. On the other hand, if \(| A| \geq 4\) \textit{N. Alon} et al. [Isr. J. Math. 188, 353--384 (2012; Zbl 1288.05124)] observed, assuming a deep Bombieri-Lang conjecture on curves of genus \(g > 1\), that the size of \(| B| \) is uniformly bounded (cf. also \textit{J. Solymosi} [Additive combinatorics, CRM Proc. Lect. Notes 43, 29--38 (2007; Zbl 1133.05096)]). The authors prove the following results: (1) There exists a set \(A\) of 3 positive integers and a corresponding set \(B \subset [0,N]\) with \(| B| \gg (\log N)^{15/17}\), such that all elements of \(A+B\) are perfect squares. (2) There exists a set \(A\) of 3 integers and a corresponding set \(B \subset[0,N]\) with \(| B| \gg (\log N)^{9/11}\), such that all elements of the sets \(A\), \(B\) and \(A + B\) are perfect squares. The proofs make use of suitably constructed elliptic curves of high rank. The first result improves the exponent \(5/7\) proved by \textit{N. Alon} et al. (loc. cit.)] for the case \(B\subset [-N,N]\).
scientific article; zbMATH DE number 6385259