Gaps between quadratic forms

Abstract

Let $△$ 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 $S(△,{\square }_2,a)$ be the set of integers $n$ such that $n\in △$, and $n+a\in {\square }_2$. We conduct a census of $S(△,{\square }_2,a)$ 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{ü}dern \& Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of M{ü}ller (1989).

19 pages