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notes/ANS Theory.md
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<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.
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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.
# Arithmetic Coding
Arithmetic coding works