Distribution Modulo One and normal numbers
Hynynen, Johanna (2025-06-10)
J. Hynynen
10.06.2025
© 2025 Johanna Hynynen. Ellei toisin mainita, uudelleenkäyttö on sallittu Creative Commons Attribution 4.0 International (CC-BY 4.0) -lisenssillä (https://creativecommons.org/licenses/by/4.0/). Uudelleenkäyttö on sallittua edellyttäen, että lähde mainitaan asianmukaisesti ja mahdolliset muutokset merkitään. Sellaisten osien käyttö tai jäljentäminen, jotka eivät ole tekijän tai tekijöiden omaisuutta, saattaa edellyttää lupaa suoraan asianomaisilta oikeudenhaltijoilta.
Julkaisun pysyvä osoite on
https://urn.fi/URN:NBN:fi:oulu-202506104291
https://urn.fi/URN:NBN:fi:oulu-202506104291
Tiivistelmä
In this Pro Gradu, based on chapters 1 and 4 of Yann Bugeaud's book Distribution Modulo One and Diophantine Approximation, we take a look at uniform distribution modulo one, normal numbers, and how they are connected.
A sequence is called uniformly distributed modulo one, if, when looking at the decimal portions of the different members of the sequence, we are just as likely to find any one number on the interval [0, 1[ as we are to find some other. Essentially, the decimal portions of the members are equally distributed on the interval [0, 1[.
A number is called simply normal to a base b, if when looking at its b-nary expansion, each of the digits {0, 1, . . . , b − 1} occurs at the same frequency. It is called normal, if it is simply normal to all bases b, b^2, b^3, . . .. Essentially, if for every possible length of a block of digits, each block of that length occurs at the same frequency.
One might see some similarity between these definitions, as the blocks in a normal number occurring at the same frequency could be described as them being equally distributed in some way. Indeed, we will find out that a number ξ is normal to a base b if and only if the sequence (ξb^n)n≥1 is uniformly distributed modulo one. With this information, the definition of normality can be expanded beyond integral bases.
Additionally, we take a look at some concepts similar to these, like the concept of the discrepancy of a sequence, and the block complexity and richness of number.
A sequence is called uniformly distributed modulo one, if, when looking at the decimal portions of the different members of the sequence, we are just as likely to find any one number on the interval [0, 1[ as we are to find some other. Essentially, the decimal portions of the members are equally distributed on the interval [0, 1[.
A number is called simply normal to a base b, if when looking at its b-nary expansion, each of the digits {0, 1, . . . , b − 1} occurs at the same frequency. It is called normal, if it is simply normal to all bases b, b^2, b^3, . . .. Essentially, if for every possible length of a block of digits, each block of that length occurs at the same frequency.
One might see some similarity between these definitions, as the blocks in a normal number occurring at the same frequency could be described as them being equally distributed in some way. Indeed, we will find out that a number ξ is normal to a base b if and only if the sequence (ξb^n)n≥1 is uniformly distributed modulo one. With this information, the definition of normality can be expanded beyond integral bases.
Additionally, we take a look at some concepts similar to these, like the concept of the discrepancy of a sequence, and the block complexity and richness of number.
Kokoelmat
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