A note on the sum-product problem for fractal sets
Fuente:
arXiv
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| Autori principali: | , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866918464795443200 |
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| author | Cushman, Adam O'Regan, William |
| author_facet | Cushman, Adam O'Regan, William |
| contents | Utilising recent advances in incidence geometry for balls and tubes, and advances in sum-product theory in the discrete setting, we show that for $0 < s \leq 1/2$ and for any $A \subset \mathbb{R}$ with Hausdorff dimension $s$, either the upper-box dimension of $AA$, or the lower-box dimension of $A+A$ must be at least $29s/23$. We obtain the slightly better bound of $33 s / 26$ when we replace the sum-set with the smoother difference-set. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_21949 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A note on the sum-product problem for fractal sets Cushman, Adam O'Regan, William Classical Analysis and ODEs Combinatorics Utilising recent advances in incidence geometry for balls and tubes, and advances in sum-product theory in the discrete setting, we show that for $0 < s \leq 1/2$ and for any $A \subset \mathbb{R}$ with Hausdorff dimension $s$, either the upper-box dimension of $AA$, or the lower-box dimension of $A+A$ must be at least $29s/23$. We obtain the slightly better bound of $33 s / 26$ when we replace the sum-set with the smoother difference-set. |
| title | A note on the sum-product problem for fractal sets |
| topic | Classical Analysis and ODEs Combinatorics |
| url | https://arxiv.org/abs/2604.21949 |