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| Format: | Preprint |
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2024
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| Online Access: | https://arxiv.org/abs/2404.01069 |
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| _version_ | 1866913293714587648 |
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| author | Iyer, Siddharth |
| author_facet | Iyer, Siddharth |
| contents | We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers $1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\| \sum_{j=1}^{k} \sqrt{a_j} \right\| = O(n^{-k/2}). \end{align*} The exponent $k/2$ improves upon the previous exponent of $c k^{1/3}$ of Steinerberger (2024), where $c>0$ is an absolute constant. We also show that for $α\in \mathbb{R}$, there exist integers $1 \leq b_1, \dots, b_k \leq n$ such that: \begin{align*} \left\| \sum_{j=1}^k \sqrt{b_j} - α\right\| = O(n^{-γ_k}), \end{align*} where $γ_k \geq \frac{k-1}{4}$ and $γ_k = k/2$ when $k=2^m - 1$, $m=1,2,\dots$. Importantly, our approach avoids the use of exponential sums. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_01069 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Distribution of sums of square roots modulo $1$ Iyer, Siddharth Number Theory 11J71 We improve upon a result of Steinerberger (2024) by demonstrating that for any fixed $k \in \mathbb{N}$ and sufficiently large $n$, there exist integers $1 \leq a_1, \dots, a_k \leq n$ satisfying: \begin{align*} 0 < \left\| \sum_{j=1}^{k} \sqrt{a_j} \right\| = O(n^{-k/2}). \end{align*} The exponent $k/2$ improves upon the previous exponent of $c k^{1/3}$ of Steinerberger (2024), where $c>0$ is an absolute constant. We also show that for $α\in \mathbb{R}$, there exist integers $1 \leq b_1, \dots, b_k \leq n$ such that: \begin{align*} \left\| \sum_{j=1}^k \sqrt{b_j} - α\right\| = O(n^{-γ_k}), \end{align*} where $γ_k \geq \frac{k-1}{4}$ and $γ_k = k/2$ when $k=2^m - 1$, $m=1,2,\dots$. Importantly, our approach avoids the use of exponential sums. |
| title | Distribution of sums of square roots modulo $1$ |
| topic | Number Theory 11J71 |
| url | https://arxiv.org/abs/2404.01069 |