Large gaps between sums of two squares

Let $\mathcal S=\{s_1<s_2<s_3<\ldots\}$ be the sequence of all natural numbers which can be represented as a sum of two squares of integers. For $X\ge2$ we denote by $g(X)$ the largest gap between consecutive elements of $\mathcal S$ that do not exceed $X$. We prove that for $X \to +\infty$ the lower bound $g(X)\geq \left(\frac{390}{449}-o(1)\right)\ln X$ holds. This estimate is twice the recent estimate by R. Dietmann and C. Elsholtz.

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