A Note on the Sum-Product Problem and the Convex Sumset Problem
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arXiv
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866908798902337536 |
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| author | Cushman, Adam |
| author_facet | Cushman, Adam |
| contents | We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[
\max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_ε \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - ε} .\] We also provide new exponents for $A \subset \mathbb{R} $ finite and convex, namely \[
\left\lvert A+A \right\rvert \gg_ε \left\lvert A \right\rvert ^{\frac{46}{29} - ε}, \] and \[
\left\lvert A-A \right\rvert \gg_ε \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -ε} .\] |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_13849 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Note on the Sum-Product Problem and the Convex Sumset Problem Cushman, Adam Combinatorics Number Theory We provide a new exponent for the Sum-Product conjecture on $\mathbb{R} $. Namely for $A \subset \mathbb{R}$ finite, \[ \max \left\{ \left\lvert A+A \right\rvert , \left\lvert AA \right\rvert \right\} \gg_ε \left\lvert A \right\rvert ^{\frac{4}{3} + \frac{10}{4407} - ε} .\] We also provide new exponents for $A \subset \mathbb{R} $ finite and convex, namely \[ \left\lvert A+A \right\rvert \gg_ε \left\lvert A \right\rvert ^{\frac{46}{29} - ε}, \] and \[ \left\lvert A-A \right\rvert \gg_ε \left\lvert A \right\rvert ^{\frac{8}{5} + \frac{1}{3440} -ε} .\] |
| title | A Note on the Sum-Product Problem and the Convex Sumset Problem |
| topic | Combinatorics Number Theory |
| url | https://arxiv.org/abs/2512.13849 |