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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2505.23428 |
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| _version_ | 1866909047942283264 |
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| author | Iyer, Siddharth |
| author_facet | Iyer, Siddharth |
| contents | 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{ü}dern \& Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of M{ü}ller (1989). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2505_23428 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Gaps between quadratic forms Iyer, Siddharth Number Theory 11N56 (Primary), 11B25, 11B34, 11B05 (Secondary) 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{ü}dern \& Dietmann (2009). Our work extends a classical result of Estermann (1932) and builds upon work of M{ü}ller (1989). |
| title | Gaps between quadratic forms |
| topic | Number Theory 11N56 (Primary), 11B25, 11B34, 11B05 (Secondary) |
| url | https://arxiv.org/abs/2505.23428 |