Large gaps between sums of two squares

Abstract

Let $\mathcal{S}=\{s_1<s_2<s_3<\dots \}$ be the sequence of all natural numbers which can be represented as a sum of two squares of integers. For $X\ge 2$ 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 $$g(X)\ge \left(\frac{390}{449}-o(1)\right)\ln X$$ holds. This estimate is twice the recent estimate by R. Dietmann and C. Elsholtz.

5 pages; This version will not be published in this form. It will appear in a joint and expanded manuscript by R. Dietmann, C. Elsholtz, A. Kalmynin, S. Konyagin and J. Maynard