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).
Categories: math.NT