Some remarks on Goldbach's problem
It is well-known that there exist sparse sets of integers \({\mathcal A}\) such that the equation \(n=a+p\) is soluble for all sufficiently large \(n\), where \(a\in{\mathcal A}\) and \(p\in{\mathcal P}\), the set of all prime numbers. Refining combinatorial methods due to \textit{G. G. Lorentz} [Proc. Am. Math. Soc. 5, 838-841 (1954; Zbl 0056.039)] and \textit{P. Erdős} [Proc. Am. Math. Soc. 5, 847-853 (1954; Zbl 0056.270)], the authors prove that for all \(A>0\) there exists a set \({\mathcal P}'\subseteq {\mathcal P}\) such that \[ |\{p\leq x,\;p\in{\mathcal P}'\} |\ll_ A (\log x)^ 2, \] for all even numbers \(n\leq x\) with at most \(O(x(\log x)^{-A})\) exceptions, the equation \(n=p+ p'\) can be solved with \(p\in{\mathcal P}\) and \(p'\in{\mathcal P}'\), and for all sufficiently large odd \(n\) the equation \(n=p_ 1+ p_ 2+ p'\) can be solved with \(p_ 1\), \(p_ 2\in{\mathcal P}\) and \(p'\in{\mathcal P}'\). Following, \textit{H. L. Montgomery} and \textit{R. C. Vaughan} [Acta Arith. 27, 353-370 (1975; Zbl 0301.10043)], the exceptional set is improved to \(O(x^{1- \delta})\) for some \(\delta>0\), at the cost of having a larger set \({\mathcal P}'\), satisfying \[ |\{p\leq x,\;p\in{\mathcal P}'\}| \ll_ \varepsilon x^ \varepsilon. \] The ternary result is also improved by showing the existence of a set \({\mathcal P}' \subseteq{\mathcal P}\) such that \[ |\{p\leq x,\;p\in{\mathcal P}'\}| \ll\log \log x. \] The proofs depend on lower bounds for the number of Goldbach representations of almost all even numbers in short intervals.
number of Goldbach representations of almost all even numbers
scientific article; zbMATH DE number 650850