On additive decompositions of the set of quadratic residues modulo \(p\) (Q2919665)

A finite or infinite subset of non-negative integers \(\mathbb{Z}_{\ge 0}\) is called primitive if it cannot be written in a nontrivial way as a sumset of two subsets of \(\mathbb{Z}_{\ge 0}\). In the opposite case it is called reducible. Further, two subsets of \(\mathbb{Z}_{\ge 0}\) are called asymptotically equal if they are equal apart from a finite number of their elements. An infinite primitive set is called totally primitive if every set asymptotically equal to it is primitive. These notions go back to \textit{H. H. Ostmann} [Additive Zahlentheorie. Erster Teil. Berlin: Springer (1956; Zbl 0072.03102), (1968; Zbl 0179.06903), p.~13] who posed the question -- later called as the ``inverse Goldbach conjecture'' -- whether the set of prime numbers is totally primitive. Despite several interesting and remarkable partial results towards its solution the conjecture as such is still open.NEWLINENEWLINEIn the paper under review the author proposes to turn the attention to finite analogues of problems like reducibility or primitivity. These notions can be naturally extended to any additive group, typically the subsets of finite fields \(\mathbb{F}_q\) (the total primitivity cannot be adapted for finite sets). This idea additionally also opens the way to new methodological approaches to the investigation of these and related problems, e.g. applications of Weil's theorem on summing characters of the values of polynomials over finite fields or results connected with the estimations of the cardinalities of sumsets in abelian groups. Both find applications in the paper.NEWLINENEWLINEAs the title of the paper indicates the subject of the paper is the question of reducibility of the set of quadratic residues modulo a prime \(p\). More precisely, the paper is devoted to the study of the following author's conjecture: NEWLINENEWLINEThe set \(\mathcal{Q}=\mathcal{Q}(p)\) of quadratic residues modulo a prime \(p\) is primitive for \(p\) sufficiently large. NEWLINENEWLINEIn the first of two theorems stated in the paper, the author proves upper and lower estimates for the minimal and maximal cardinality of the summands \(\mathcal{U},\mathcal{V}\) in a 2-decomposition \(\mathcal{Q}=\mathcal{U}+\mathcal{V}\) (if there is any). Based on these estimates he then proves in the second theorem that the Elsholtz-type result of a 3-decomposition of \(\mathcal{Q}\) is impossible for primes \(p\) large enough. NEWLINENEWLINEIn the closing remarks he then notices that both results can be extended to finite fields. However an extension to structures like \(\mathbb{Z}/m\mathbb{Z}\) can be much more complicated due to the problems with the extension of Weil's theorem for the case of composite moduli.