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notes/ANS Theory.md
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<https://arxiv.org/abs/1311.2540> <https://arxiv.org/abs/1311.2540>
In standard numeral systems, different digits are treated as containing the same amount of information. A 7 stores the same amount of info as a 9, which stores the same amount of info as a 1. In standard numeral systems, different digits are treated as containing the same amount of information. A 7 is stored using the same amount of info as a 9, which is stored using the same amount of info as a 1, that is, 1 digit.
This makes the amount of information a single digit stores *uniform* across all digits. However, that's far from the most efficient way to represent most datasets. This makes the amount of information a single digit stores *uniform* across all digits. However, that's far from the most efficient way to represent most datasets, because real world data rarely follows a uniform distribution.
ANS theory is based around the idea that digits that occur more often can be stored in a way that requires less information, and digits that occur less often can be stored using more information. ANS theory is based around the idea that digits that occur more often can be stored in a way that requires less information, and digits that occur less often can be stored using more information.
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Given that $x$ represents a natural number, and $s$ is the digit we're adding. In a standard binary system, adding $s$ to the least significant position means that in the new number $x$ (before the addition) now represents the Nth appearance of an even (when $s = 0$ ), or odd (when $s = 1$). With ANS, the goal is is to make that asymmetrical, so that you can represent more common values with a denser representation. Given that $x$ represents a natural number, and $s$ is the digit we're adding. In a standard binary system, adding $s$ to the least significant position means that in the new number $x$ (before the addition) now represents the Nth appearance of an even (when $s = 0$ ), or odd (when $s = 1$). With ANS, the goal is is to make that asymmetrical, so that you can represent more common values with a denser representation.
# Arithmetic Coding
Arithmetic coding works