Mysterious Numbers from Gold List Approximants

An approximant is the best fractional approximation to a target for a given denominator. (See the Details section for a precise definition and an example.) Keeping only the approximants that are better than all the previous ones produces the gold list. The gold list of approximants of a real number can exhibit fascinating patterns.
This Demonstration lets you examine the gold lists of 12 different numbers, including irrationals like and , and two rational targets. You can choose to see the error terms, which decrease in absolute value. But the "mystery" is when we (separately) subtract numerators and denominators of successive gold list entries, as shown in the last column. Amazingly enough, most of these appeared previously in the gold list itself!

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DETAILS

An approximant of a real number is defined to be a rational approximation in which the numerator is the best choice for the given denominator. For example, the list of all approximants of starts off this way: . For each of the denominators, the numerator is chosen so as to approximate as closely as possible.
The gold list is formed by keeping only approximants that successively approach the target more closely. In the case of , many of the approximants listed above must be eliminated: and are equal to , not closer to . Similarly, all approximants having denominators between 8 and 56 are eliminated because none of them are closer to than . The gold list for starts .
Clearly, irrational targets have infinite gold lists. None of the approximants are equal to the target, but as we consider larger denominators, closer approximants can always eventually be found. If the target itself is rational, however, its gold list terminates with the fraction itself. For example, the gold list of is precisely . This coincides with the first five elements of the gold list of , but then terminates since there are no fractions closer to the rational target than the fraction itself.
Now, in a move that seems unmotivated and arbitrary, we consider the differences between numerators and denominators of successive gold list entries, referred to here as and , respectively. These are shown in the last column of the table.
Surprisingly, the ratios are almost always themselves earlier members of the gold list, shown in the table using the same color for convenient identification. For the gold list of , the sequence of these "delta ratios" starts this way: . In fact, except for occasional intrusions, we recognize all these new ratios. Initially there seems to be no reason for this mysterious behavior.
The delta ratios sequence is formed by pairing successive gold list elements and "subtracting"—actually subtracting numerators and denominators separately. The inverse operation of "adding" previous elements of the gold list provides useful insights. First, it must be noted that this operation is not "incorrect" addition of fractions, but rather forming the mediant, denoted here by the symbol "⊕". The mediant of fractions and is defined as . Now most elements of the gold list of can be seen to be mediants of previous elements. A further definition allows all but the first pair to be written in terms of earlier approximants. The weighted mediant of fractions and with weighting factor is defined as . Using this notation, several subsequences of weighted mediants are revealed in the gold list of . For example,
and .
Subsequences where the same approximants are used repeatedly to form successive gold list entries always approach the target monotonically from one side or the other, as indicated by the error terms. These repeated mediant subsequences have the form
,
where and are themselves approximants on opposite sides of the target. The through mediants are all on the side of the target, approaching it, until the mediant has crossed over to the side.
To allow real-time interactivity, the code used here relies on the ultra-rapid iterative bracket method of generating the gold list, rather than the painfully slow method based on the gold list definition: testing each approximant for all denominators and either keeping or discarding them. Additional material relating to approximants, mediants, weighted or repeated mediants, and methods for generating the gold list can be found in [1].
Snapshot 1: The beginning of the gold list of , containing familiar approximations such as , and . These also show up later in the column of numerator and denominator differences, meaning that most gold list entries are actually mediants of earlier approximants. Some of the best approximants (colored in the first column) are used in sequences of repeated mediants, approaching the target from one side or the other. The strikingly precise is used in a sequence of almost 150 repeated mediants approaching from below.
Snapshot 2: The beginning of the gold list of the golden ratio , in which every gold list entry appears exactly once in the column of numerator and denominator differences, a situation closely related to the continued fraction expansion of the target.
Snapshot 3: The beginning of the gold list of , itself a good approximation to , having an identical gold list up to the point where the rational target is reached.
Reference
[1] R. L. Caviness and K. E. Caviness. Making Pi(e) from Scratch, Rapidly and Memorably [Video]. AMATYC Webinar (Sep 27, 2019) www.youtube.com/watch?v=JcBr_v-n6AA.
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