A construction of a $λ$- Poisson generic sequence
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arXiv
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| Format: | Preprint |
| Publié: |
2022
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| _version_ | 1866913245569220608 |
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| author | Becher, Verónica Himelfarb, Gabriel Sac |
| author_facet | Becher, Verónica Himelfarb, Gabriel Sac |
| contents | Years ago Zeev Rudnick defined the $λ$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $λ$. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $λ$-Poisson generic sequence over any alphabet and any positive $λ$, except for the case of the two-symbol alphabet, in which it is required that $λ$ be less than or equal to the natural logarithm of $2$. Since $λ$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $λ$-Poisson generic. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2205_03981 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A construction of a $λ$- Poisson generic sequence Becher, Verónica Himelfarb, Gabriel Sac Number Theory Discrete Mathematics 11K16, 05A05, 60G55 Years ago Zeev Rudnick defined the $λ$-Poisson generic sequences as the infinite sequences of symbols in a finite alphabet where the number of occurrences of long words in the initial segments follow the Poisson distribution with parameter $λ$. Although almost all sequences, with respect to the uniform measure, are Poisson generic, no explicit instance has yet been given. In this note we give a construction of an explicit $λ$-Poisson generic sequence over any alphabet and any positive $λ$, except for the case of the two-symbol alphabet, in which it is required that $λ$ be less than or equal to the natural logarithm of $2$. Since $λ$-Poisson genericity implies Borel normality, the constructed sequences are Borel normal. The same construction provides explicit instances of Borel normal sequences that are not $λ$-Poisson generic. |
| title | A construction of a $λ$- Poisson generic sequence |
| topic | Number Theory Discrete Mathematics 11K16, 05A05, 60G55 |
| url | https://arxiv.org/abs/2205.03981 |