Gaps in Multiplicative Sidon Sets
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arXiv
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| Auteurs principaux: | , , |
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| Format: | Preprint |
| Publié: |
2026
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| _version_ | 1866910187772706816 |
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| author | van Doorn, Wouter Monticone, Pietro Tang, Quanyu |
| author_facet | van Doorn, Wouter Monticone, Pietro Tang, Quanyu |
| contents | For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\subseteq\{1,2,\dots,n\}$ that intersects every interval $[x,x+L]\subseteq[1,n]$. Sárközy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\le\sqrt n$ for every $n\in\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve the upper bound further and, with $ρ= \frac{13-\sqrt{69}}{10} < 0.47$, prove that $g(n)\ll_{\varepsilon} n^{ρ+\varepsilon}$ for every $\varepsilon > 0$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_02064 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gaps in Multiplicative Sidon Sets van Doorn, Wouter Monticone, Pietro Tang, Quanyu Number Theory Combinatorics 11B75, 11B05, 11B83, 11N05, 11N36, 05D15, 68V20 For a positive integer $n$, let $g(n)$ denote the infimum of all real numbers $L$ such that there exists a multiplicative Sidon set $A\subseteq\{1,2,\dots,n\}$ that intersects every interval $[x,x+L]\subseteq[1,n]$. Sárközy asked for estimates on $g(n)$, and he in particular asked whether one has $g(n)\le\sqrt n$ for every $n\in\mathbb{N}$. We first show that this estimate does indeed hold, with a proof that was autonomously discovered and formally verified in Lean by Aristotle. Next, we improve the upper bound further and, with $ρ= \frac{13-\sqrt{69}}{10} < 0.47$, prove that $g(n)\ll_{\varepsilon} n^{ρ+\varepsilon}$ for every $\varepsilon > 0$. |
| title | Gaps in Multiplicative Sidon Sets |
| topic | Number Theory Combinatorics 11B75, 11B05, 11B83, 11N05, 11N36, 05D15, 68V20 |
| url | https://arxiv.org/abs/2605.02064 |