Let \(E(X,H)\) denote the number of even integers \(n\) with \(X<n\leq X+H\) such that \(n\) cannot be written as a sum of two primes, and set \(H=X^{\theta}\). It is known that \(E(X,H)=o(H)\) as \(X\rightarrow\infty\), if \(\theta\) is larger than a certain small number. In fact, G. Harman showed that if \(\theta\geq11/180\), then for any fixed \(A\), one has \(E(X,H)\ll H(\log X)^{-A}\) (see Chapter 10 of [Prime-detecting sieves. London Mathematical Society Monographs 33. Princeton, NJ: Princeton University Press (2007; Zbl 1220.11118)]).
Meanwhile, the methods of H. L. Montgomery and R. C. Vaughan enabled us to establish a sharper bound of the form \(E(X,H)\ll H^{1-\delta}\) with some constant \(\delta>0\), at least when \(H\) is as large as \(X\). Several mathematicians have worked to show bounds of the latter type for smaller \(H\), and for example T. P. Peneva [Monatsh. Math. 132, No. 1, 49–65 (2001; Zbl 0974.11037), Corrigendum 141, No. 3, 209–217 (2004; Zbl 1111.11312)] and A. Languasco [Monatsh. Math. 141, No. 2, 147–169 (2004; Zbl 1059.11059)] showed such bounds, respectively, for \(\theta>1/3\) and \(\theta>7/24\). And in the paper under review, the author proves that for \(\theta>1/5\), one has \(E(X,H)\ll H^{1-\delta}\) with some effectively computable positive number \(\delta\) depending on \(\theta\), substantially improving previous results in this direction. (The author points out, and overcomes, a flaw in the argument of Languasco (loc. cit.).) It may be said in brief that the proof is constructed by incorporating ideas of sieve methods into the previous work being based on the circle method.