Gaps between quadratic forms

TL;DR

This paper investigates the distribution of integers that are represented by a specific quadratic form and also as sums of two squares, providing lower bounds on their density in short intervals.

Contribution

It extends classical results by establishing lower bounds for the count of such integers in short intervals using advanced number theory techniques.

Findings

Existence of a constant H_a ensuring a lower bound on the count in short intervals.

Application of Tolev's theorem on sums of two squares in arithmetic progressions.

Analysis of a multiplicative function related to the problem.

Abstract

Let $\triangle$ denote the integers represented by the quadratic form $x^2+xy+y^2$ and $\square_{2}$ denote the numbers represented as a sum of two squares. For a non-zero integer $a$, let $S(\triangle,\square_{2},a)$ be the set of integers $n$ such that $n \in \triangle$, and $n + a \in \square_{2}$. We conduct a census of $S(\triangle,\square_{2},a)$ in short intervals by showing that there exists a constant $H_{a} > 0$ with \begin{align*} \# S(\triangle,\square_{2},a)\cap [x,x+H_{a}\cdot x^{5/6}\cdot \log^{19}x] \geq x^{5/6-\varepsilon} \end{align*} for large $x$. To derive this result and its generalization, we utilize a theorem of Tolev (2012) on sums of two squares in arithmetic progressions and analyse the behavior of a multiplicative function found in Blomer, Br{\"u}dern \& Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of M{\"u}ller (1989).