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I can’t believe that I’m the first to follow this train of thought, since each step is obvious. Therefore, I am tagging this as a reference request. Someone please quote me chapter and verse where this thread has been dealt with before. Thanks.

Step 1. Suppose x > 1. Hmm, x might be really big, but we want to get a handle on x, to downsize x, without losing, so to speak, the essential properties of x. How can we do this?

Step 2. The Harmonic Sequence leaps to mind. Because of the divergence of the corresponding series (i.e., the Harmonic Series), if we start subtracting from x the terms of the Harmonic Sequence, it will be after only finitely many such subtractions that we exhaust x, not matter how big x was.

Step 3. Define H(n) to be the sum of the first n terms of the Harmonic Sequence, for each positive integer n. So, H(n) = 1/1 + 1/2 + … + 1/n.

Step 4. So, there is a unique positive integer n such that H(n) ≤ x < H(n + 1). Let us call that positive integer n the index of x, denoting it by index(x), and let us call the quantity x – H(n) the residue of x, denoting it by r(x).

Commentary. The residue of x might already by a useful quantity. Whether it is in fact so is what the title of this post is asking.

For example, is there a useful characterization of r(xy) in terms of r(x) and r(y), somewhat along the lines of characterizing the logarithm? Perhaps. Or perhaps to get non-trivial results, we must, as so often happens, enter into iteration. So, how can we continue the process, in a way that ties back to x in an essential way? Again, the answer is obvious.

Step 5. Consider the quantity H(n + 1) – x, where n is the index of x. Let’s call this the complementary residue of x, denoting it by c(x). It is obvious that 0 < c(x) <= H(n + 1). How can we best use c(x)? Again, the answer is obvious: take its reciprocal.

Step 6. For x > 1, define the sphinx of x to be 1/c(x), denoting it by sphinx(x). Note that for really large x, the sphinx of x will also be really large.

Step 7. For a given x > 1, set up the following 5-fold sequence:

pathx(1) = x

pathindex(1) = index(pathx(1))

pathr(1) = r(pathx(1))

pathc(1) = c(pathx(1))

pathsphinx(1) = 1/pathc(1)

For each positive integer n > 1:

pathx(n) = pathsphinx(n – 1)

pathindex(n) = index(pathx(n))

pathr(n) = r(pathx(n))

pathc(n) = c(pathx(n))

pathsphinx(n) = 1/pathc(n)

Step 8. For x > 1, define the distillation sequence of x to be the sequence {y(n)} of positive integers such that for each positive integer n, y(n) = pathindex(n).

The obvious question is then whethere, for a given x > 1, the distillation sequence of x characterizes x in any interesting or useful way. For example, x is algebraic fif its distillation sequence has such-and-such a property.

Regards,

Mike Jones

28.May.2011 (Beijing time)

Added by OP on May 30, 2011 (Beijing time)

My motivation for the original investigation along these lines was to solve the open problem of finding an explicit well-ordering for the set of real numbers. My strategy was simply that I would associate a unique sequence of positive integers with each real number, and then let the well-ordering of the set of positive integers induce a well-ordering on the set of real numbers. Since finding an explicit well-ordering for any non-empty connected subset of the reals is equivalent to finding an explicit well-ordering for the reals, it seemed apropos to deal with (1,infinity). So, this was the motivation for step 1, and the harvesting, so to speak, of the distillation sequence. However, it seems to me that this sequence is not unique for a given x > 1. However, the infinite matrix whose first row is the distillation sequence for x, whose second row is the distillation sequence for x + 1, and whose third row is the distillation sequence for x + 2, and so on MIGHT be uniquely defined by x. I call this matrix the Serengeti matrix for x. But we need a single unique sequence for x. Using the diagonal sequence of course suggests itself. I call this the Serengeti sequence for x. If the Serengeti sequence is unique to x, then that solves the problem, but I have not had the time/ability to determine this. But I thought that the partial results along the way, such as the distillation sequence possibly yielding useful information was of interest in its own right, but since high-rep users are balking at step 1, I guess I need to give a full account of my motivation, and so now you have it:-)

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    @Mike Jones: Have never seen this before. Note that if $x$ is big, then $n$ is humongous, but at least an integer. The bigger $x$ is, the more precisely $n$ measures $x$, which sounds like the wrong direction. In Step 5, didn't you want to say $c(x) \le 1/(n+1)$?2011-05-28
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    @Mike Jones: I'm afraid I don't find any of these steps or concepts "obvious"; I don't see any reason why they would be useful or satisfy any nice properties. We could do the same thing with any divergent series - there's nothing special here about the harmonic series. Furthermore, as user6312 has pointed out, your process makes bigger numbers, not smaller ones. But at any rate, I highly doubt that anyone has investigated this before.2011-05-28
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    Zev, I can see that it is well-motivated at least up to Step 5. H(x*y) ~ H(x)+H(y)+gamma is true in the limit. I am having trouble following the rest myself and it would be helpful to me if Mike could express it in some way other than pseudocode. Also I'm not familiar with the terms "sphinx" or "downsizing" and this may be why I'm having trouble following the intention of the pseudocode. Mike, can you clarify these aspects?2011-05-28
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    @Dan: Okay, I should have been more specific. Regarding Step 1: Firstly, here are some functions that "downsize" numbers nicely: for any "small" $a<1$ of your choice, $$x\mapsto ax,$$ $$x\mapsto x^{a},$$ both of which I find much more "obvious" than Mike's construction. Secondly, the idea of "downsizing" makes no sense, because all real numbers are "equally big" (for example, just because we *conceptualize* $10^{-10000}$ as "small" doesn't mean anything). Any choice of definition of "downsizing" will produce numbers that aren't any easier or harder to "get a handle on" than the original.2011-05-28
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    Zev, I'm not sure what Mike means by "downsizing" but I don't think it's anything as simple as a decreasing function. He suggests that there is a process for determining if e.g. x is algebraic.2011-05-28
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    Regarding Step 2: Any divergent series has this property, so the choice of the harmonic series is not "obvious" in any way. Regarding Steps 4, 5 and 6: Since the arbitrary choice of the harmonic series wasn't obvious, none of these definitions are obvious either, especially the ones in Steps 5 and 6. I am unable to parse Steps 7 and 8 (I agree, Dan, the pseudocode is not clear), but if it were really as "obvious" as suggested I think there would not be such an issue.2011-05-28
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    @Dan: I can define any algorithm and suggest that it might characterize \_\_\_\_\_. Short of actually determining whether it does so or not (for which I have none of the necessary time, motivation, or understanding of Steps 7 and 8), I would judge such a suggestion on the basis of whether there is *any reason to think that it does*, and in my opinion, the answer in this case is no.2011-05-28
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    @Zev Chonoles The algorithm here is essentially doing a continued fraction expansion, except instead of the terms in the expansion being integers, they are harmonic numbers which are increasing at an alarmingly fast rate. There are interesting and useful things that come from continued fraction expansions (e.g. repetition means you are of the form $a+b\sqrt{c}$). This variant seems much more arbitrarily and much harder to analyze.2011-05-28
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    Aaron, it appears that little can be said in general about whether or not a continued fraction is algebraic. Maybe you can use this in your answer? http://modular.math.washington.edu/edu/2007/spring/ent/ent-html/node73.html2011-05-28
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    @Aaron: I am familiar with the stated result about continued fraction expansions and real quadratic irrationals. However, if there is any reason to think Mike's definition will have some relation to the usual continued fraction expansion, then either this reason is independent of the choice of using the harmonic series, and would work equally well with any other divergent series (which seems unlikely to me), or it depends somehow on using the harmonic series and not some other one, and no such reason is apparent to me. In other words: I agree, Mike's definition is arbitrary.2011-05-28
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    Also, I appreciate your explanation of what is going on in Steps 7 and 8.2011-05-28
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    I don't understand Step 1. You want to compress a number so as to keep its "essential properties." What are these essential properties? The properties you want to preserve inform what kind of compression you do.2011-05-28
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    @user6312: Yes, you're right about that typo. I have edited the post, but was unable to drop in the proper symbol, so used "<=". Thanks.2011-05-28
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    @Dan Brumbleve: I did not use "downsizing" in the technical portion, only as an intuitive tag for what I thought was obvous: r(x) is much less than x. Since I don't yet have the reference that I am requesting, I have to make up my own terminology to do this presention. I thought that the term "sphinx" was appropriate, because that is how x is "re-born" for another go-around:-)2011-05-28
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    @Qiaochu Yuan: I don't know either what the essential properties are, but I believe that most of them are left intact when a rational number is subtracted from the number in question:-)2011-05-28
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    @Mike: I don't want to quash your enthusiasm here, but this is not how you go about creating useful mathematical concepts. There's no _a priori_ reason that this transformation should have any reasonable properties or be useful in any application.2011-05-28
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    +1 Qioachu, couldn't have said it better. Also, @Mike, what do you think of my suggestions for "downsizing functions" in my comment above - do they meet your criteria? If not, can you articulate why not?2011-05-28
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    I just read this post and had a lot of the same difficulties as the other commenters. In particular, Step 1 makes no sense at all to me: as far as I can see the size of a positive real number *is* its essential property. I would suggest removing this step, since it adds nothing.2011-05-28
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    (cont'd) Moreover, I don't think it is a "real question" to simply give an algorithm or class of sequence and ask if it is useful for something. Figuring that out *is* the task of doing mathematical research. An acceptable modification would be simply to ask "Has anyone seen this construction before?" The only downside of this is that if the answer is "no", then it will be hard to know this definitively.2011-05-28
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    @Qiaochu Yuan: Since when does anyone know of a procedure for creating useful mathematical concepts? (That is, looking forward, not backward.)2011-08-09
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    @Mike: there are no foolproof procedures, but there are better and worse ways to search the space of possible ideas for interesting ones.2011-08-09
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    @Qiaochu Yuan: Really? And has anyone ever documented those ways?? Sounds like this would make a great Big List project if not. Anyway, this was a question that I was genuinely interested in, and that should be a sufficient condition for presenting it on MSE, right?2011-08-14

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