On the number of 3APs in fractal sets
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866914303760662528 |
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| author | Carnovale, Marc Senger, Steven |
| author_facet | Carnovale, Marc Senger, Steven |
| contents | We use techniques from the study of the Falconer distance conjecture to explore conditions which guarantee largeness (in terms of bounded $L^2$ density/Lebesgue measure and Hausdorff measure) of the set of lengths of step-sizes of three-term arithmetic progressions which occur within fractal sets, as well as analogous statements in discrete settings. Our main result is a version of Łaba and Pramanik's result in arxiv:0712.3882 that relies only on an assumption of a lower bound, $δ$, on the mass of the measure $μ$ together with an upper bound, $M$ on the $L^q$ norm of its Fourier transform for some $q\in(2,3]$ depending on the parameters $δ$ and $M$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2602_03029 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On the number of 3APs in fractal sets Carnovale, Marc Senger, Steven Classical Analysis and ODEs 28A80, 11B25, 28A75 We use techniques from the study of the Falconer distance conjecture to explore conditions which guarantee largeness (in terms of bounded $L^2$ density/Lebesgue measure and Hausdorff measure) of the set of lengths of step-sizes of three-term arithmetic progressions which occur within fractal sets, as well as analogous statements in discrete settings. Our main result is a version of Łaba and Pramanik's result in arxiv:0712.3882 that relies only on an assumption of a lower bound, $δ$, on the mass of the measure $μ$ together with an upper bound, $M$ on the $L^q$ norm of its Fourier transform for some $q\in(2,3]$ depending on the parameters $δ$ and $M$. |
| title | On the number of 3APs in fractal sets |
| topic | Classical Analysis and ODEs 28A80, 11B25, 28A75 |
| url | https://arxiv.org/abs/2602.03029 |