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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.18297 |
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| _version_ | 1866913493107605504 |
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| author | Cutler, Jonathan Pebody, Luke Sarkar, Amites |
| author_facet | Cutler, Jonathan Pebody, Luke Sarkar, Amites |
| contents | Given a set of integers $A$ and an integer $k$, write $A+k\cdot A$ for the set $\{a+kb:a\in A,b\in A\}$. Hanson and Petridis showed that if $|A+A|\le K|A|$ then $|A+2\cdot A|\le K^{2.95}|A|$. At a presentation of this result, Petridis stated that the highest known value for $\frac{\log(|A+2\cdot A|/|A|)}{\log(|A+A|/|A|)}$ (bounded above by 2.95) was $\frac{\log 4}{\log 3}$. We show that, for all $ε>0$, there exist $A$ and $K$ with $|A+A|\le K|A|$ but with $|A+2\cdot A|\ge K^{2-ε}|A|$.
Further, we analyse a method of Ruzsa, and generalise it to give continuous analogues of the sizes of sumsets, differences and dilates. We apply this method to a construction of Hennecart, Robert and Yudin to prove that, for all $ε>0$, there exists a set $A$ with $|A-A|\ge |A|^{2-ε}$ but with $|A+A|<|A|^{1.7354+ε}$.
The second author would like to thank E. Papavassilopoulos for useful discussions about how to improve the efficiency of his computer searches. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_18297 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Sums, Differences and Dilates Cutler, Jonathan Pebody, Luke Sarkar, Amites Combinatorics Given a set of integers $A$ and an integer $k$, write $A+k\cdot A$ for the set $\{a+kb:a\in A,b\in A\}$. Hanson and Petridis showed that if $|A+A|\le K|A|$ then $|A+2\cdot A|\le K^{2.95}|A|$. At a presentation of this result, Petridis stated that the highest known value for $\frac{\log(|A+2\cdot A|/|A|)}{\log(|A+A|/|A|)}$ (bounded above by 2.95) was $\frac{\log 4}{\log 3}$. We show that, for all $ε>0$, there exist $A$ and $K$ with $|A+A|\le K|A|$ but with $|A+2\cdot A|\ge K^{2-ε}|A|$. Further, we analyse a method of Ruzsa, and generalise it to give continuous analogues of the sizes of sumsets, differences and dilates. We apply this method to a construction of Hennecart, Robert and Yudin to prove that, for all $ε>0$, there exists a set $A$ with $|A-A|\ge |A|^{2-ε}$ but with $|A+A|<|A|^{1.7354+ε}$. The second author would like to thank E. Papavassilopoulos for useful discussions about how to improve the efficiency of his computer searches. |
| title | Sums, Differences and Dilates |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2402.18297 |