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| Autore principale: | |
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| Soggetti: | |
| Accesso online: | https://arxiv.org/abs/2404.12279 |
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| _version_ | 1866917644697862144 |
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| author | Rees, Mary |
| author_facet | Rees, Mary |
| contents | Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under $n$ Collatz iterates, modulo $2^p$, is usefully close to uniform distribution on the integers modulo $2^p$, if $p/n$ is small enough. Conjecture 2 is that the density is bounded from zero for the incidence of both $0$ and $1$ for the coefficients in the dyadic expansions of $-3^{-\ell }$ on all but an exponentially small set of paths of a geometrically distributed random walk on the two-dimensional array of these coefficients. It is shown that Conjecture 2 implies Conjecture 1. At present, Conjecture 2 is unresolved. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_12279 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Push-forward of geometric distributions under Collatz iteration: Part 1 Rees, Mary Probability Two conjectures are presented. The first, Conjecture 1, is that the pushforward of a geometric distribution on the integers under $n$ Collatz iterates, modulo $2^p$, is usefully close to uniform distribution on the integers modulo $2^p$, if $p/n$ is small enough. Conjecture 2 is that the density is bounded from zero for the incidence of both $0$ and $1$ for the coefficients in the dyadic expansions of $-3^{-\ell }$ on all but an exponentially small set of paths of a geometrically distributed random walk on the two-dimensional array of these coefficients. It is shown that Conjecture 2 implies Conjecture 1. At present, Conjecture 2 is unresolved. |
| title | Push-forward of geometric distributions under Collatz iteration: Part 1 |
| topic | Probability |
| url | https://arxiv.org/abs/2404.12279 |