Finding large additive and multiplicative Sidon sets in sets of integers
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866913412131323904 |
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| author | Jing, Yifan Mudgal, Akshat |
| author_facet | Jing, Yifan Mudgal, Akshat |
| contents | Given $h,g \in \mathbb{N}$, we write a set $X \subset \mathbb{Z}$ to be a $B_{h}^{+}[g]$ set if for any $n \in \mathbb{Z}$, the number of solutions to the additive equation $n = x_1 + \dots + x_h$ with $x_1, \dots, x_h \in X$ is at most $g$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{h}^{\times}[g]$ set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $g \in \mathbb{N}$ and $δ>0$ such that for any $h \in \mathbb{N}$ and for any finite set $A$ of integers, the largest $B_{h}^{+}[g]$ set $B$ inside $A$ and the largest $B_{h}^{\times}[g]$ set $C$ inside $A$ satisfy \[ \max \{ |B| , |C| \} \gg_{h} |A|^{(1+ δ)/h }. \] In fact, when $h=2$, we may set $g = 31$, and when $h$ is sufficiently large, we may set $g = 1$ and $δ\gg (\log \log h)^{1/2 - o(1)}$. The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2203_13174 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Finding large additive and multiplicative Sidon sets in sets of integers Jing, Yifan Mudgal, Akshat Number Theory Combinatorics 11B30, 11B83, 05D40 Given $h,g \in \mathbb{N}$, we write a set $X \subset \mathbb{Z}$ to be a $B_{h}^{+}[g]$ set if for any $n \in \mathbb{Z}$, the number of solutions to the additive equation $n = x_1 + \dots + x_h$ with $x_1, \dots, x_h \in X$ is at most $g$, where we consider two such solutions to be the same if they differ only in the ordering of the summands. We define a multiplicative $B_{h}^{\times}[g]$ set analogously. In this paper, we prove, amongst other results, that there exist absolute constants $g \in \mathbb{N}$ and $δ>0$ such that for any $h \in \mathbb{N}$ and for any finite set $A$ of integers, the largest $B_{h}^{+}[g]$ set $B$ inside $A$ and the largest $B_{h}^{\times}[g]$ set $C$ inside $A$ satisfy \[ \max \{ |B| , |C| \} \gg_{h} |A|^{(1+ δ)/h }. \] In fact, when $h=2$, we may set $g = 31$, and when $h$ is sufficiently large, we may set $g = 1$ and $δ\gg (\log \log h)^{1/2 - o(1)}$. The former makes progress towards a recent conjecture of Klurman--Pohoata and quantitatively strengthens previous work of Shkredov. |
| title | Finding large additive and multiplicative Sidon sets in sets of integers |
| topic | Number Theory Combinatorics 11B30, 11B83, 05D40 |
| url | https://arxiv.org/abs/2203.13174 |