On the exceptional set of Goldbach numbers in short intervals

Let $E$ be the set of integers which are not a sum of two primes, $E(X)=E\cap[1,X]$ and $E(X,H)=E\cap[X,X+H]$, where $H=o(X)$. A well known result of Montgomery-Vaughan proves that there exists an absolute positive constant $\delta$ such that $\vert E(X)\vert \ll X^{1-\delta}$. Here we prove that there exists an absolute positive constant $\delta$ such that, for $H\geq X^{7/24+7\delta}$, $\vert E(X,H)\vert \ll H^{1-\delta/600}$, improving a result by Peneva.