Large gaps between sums of two squares

TL;DR

This paper establishes a new lower bound on the size of gaps between numbers representable as sums of two squares, showing they grow at least logarithmically with a specific constant, improving previous results.

Contribution

It provides a sharper lower bound on the maximal gaps between sums of two squares, nearly doubling the previous estimate by Dietmann and Elsholtz.

Findings

Lower bound g(X) ≥ (390/449 - o(1)) ln X as X→∞

Gaps between sums of two squares grow at least logarithmically

Improves previous lower bounds on these gaps

Abstract

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.