Parallelopipeds of positive rank twists of elliptic curves (Q2884652)

Let \(S\) be a set of positive integers which has positive lower density, i.e., NEWLINE\[NEWLINE{\liminf}_{x\to\infty} \frac{\#\{ z \in S : z \leq x \}}{x} >0.NEWLINE\]NEWLINE The authors of the paper under review prove that for each positive integer \(n\) there are \(n\) rational numbers \(a_1,\dots,a_n\) and a rational number \(c\) such that \(S\) contains the subset NEWLINE\[NEWLINE \{ c\cdot a_1^{r_1}\cdots a_n^{r_n} : r_j=0,\, 1\},NEWLINE\]NEWLINE and that the images of \(a_j\)'s in the \(\mathbb F_2\)-vector space \(\mathbb{Q}^*/(\mathbb{Q}^*)^2\) form a linearly independent subset. In other words, the property of lower density implies that the image of \(S\) in \(\mathbb{Q}^*/(\mathbb{Q}^*)^2\) contains a subspace of every dimension.NEWLINENEWLINEThe \(\mathbb F_2\)-vector space \(\mathbb{Q}^*/(\mathbb{Q}^*)^2\) is often used as the space of parameters \(d\) for quadratic twists of an elliptic curve \(E\) over \(\mathbb{Q}\). There are conditional and unconditional results in the literature about the lower density of parameters \(d\) for quadratic twists of an elliptic curve which have certain Mordell-Weil ranks, and the authors state as a theorem that those subsets of positive lower density possess some abundant multiplicative structure as well.