On multiplicative decompositions of the set of the shifted quadratic residues modulo \(p\) (Q2874932)

The main result is as follows: Let \(p\) be a sufficiently large prime, and let \(c\in\mathbb F_p=\mathbb Z/p\mathbb Z\) with \(c\not=0\). Define \(\Omega_c'=\{x^2+c:\;x\in \mathbb F_p^\ast\}\setminus\{0\}\). If \(A\) and \(B\) are subsets of \(\Omega_c'\) with \(1<|A|\leq|B|\), then \(|A|>\sqrt p/(3\log p)\) and \(|B|<\sqrt p\log p\).NEWLINENEWLINEThis result implies that if \(p\) is a sufficiently large prime and \(c\in\mathbb F_p\setminus\{0\}\) then \(ABC\not=\Omega_c'\) for any subsets \(|A|,|B|,|C|\) of \(\Omega_c'\) with \(\min\{|A|,|B|,|C|\}\geq2\)NEWLINENEWLINEFor the entire collection see [Zbl 1279.00053].