On the exceptional set of Goldbach's problem in short intervals (Q1879312)

Let \(X\) and \(H\) be real numbers exceeding 2, and define \(E(X,H)\) to be the number of even integers in the interval \([X, X+H]\) that cannot be written as a sum of two primes. \textit{C.-H. Jia} [Acta Arith. 77, 207--287 (1996; Zbl 0863.11066)] proved that for any given positive numbers \(A\) and \(\varepsilon\), one has \(E(X,H)\ll H(\log X)^{-A}\) provided that \(H\geq X^{7/108+\varepsilon}\). But when we would assure a bound of the sharper form \(E(X,H)\ll H^{1-\delta}\) with some fixed positive number \(\delta\), currently we must admit a stronger restriction on \(H\). In this paper, the author shows that if \(\delta\) is a sufficiently small positive number and \(H\geq X^{7/24+7\delta}\), then one has \(E(X,H)\ll H^{1-\delta/600}\). This theorem improves the previously best result in the latter direction due to \textit{T. P. Peneva} [Monatsh. Math. 132, 49--65 (2001; Zbl 0974.11037); see also ibid. 141, No. 3, 209--217 (2004; Zbl 1111.11312)] in which essentially \(1/3\) took the place of the exponent \(7/24\) above. Roughly speaking, the proof is constructed by adding the idea of the localisation trick to the above work of Peneva.