[HN Gopher] Maybe powers of p don't have unexpectedly good appro...
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Maybe powers of p don't have unexpectedly good approximations?
Author : thomasahle
Score : 51 points
Date : 2022-07-13 16:42 UTC (6 hours ago)
(HTM) web link (11011110.github.io)
(TXT) w3m dump (11011110.github.io)
| madcaptenor wrote:
| I wonder what happens for powers of e. This should be different
| than pi, since e has continued fraction coefficients that make a
| nice pattern [2, 1, 2, 1, 1, 4, 1, 1, ...]. e^2 does as well
| (https://oeis.org/A001204). (I know this fact but have never
| understood any of the proofs.)
|
| But e^3 (https://oeis.org/A058282) does not have a "nice
| pattern", and neither does e^4 (https://oeis.org/A058283)
| pyrolistical wrote:
| Often a nice pattern can be found if one uses a generalized
| continued fraction
| westurner wrote:
| /? 3blue1brown e^ipi
| https://m.youtube.com/results?sp=mAEA&search_query=3blue1bro...
|
| Given: e = limit((1 + 1/n)^n, +[?]) # Euler's
| number i = [?]-1 # orthogonal; i_0^2 = -1 pi =
| (666/212 - 22/7)*p # circle circumference / diameter
|
| Euler's identity: e^ip + 1 = 0
|
| Euler's formula: e^ix = cos(x) + i*sin(x)
|
| Euler's formula: https://en.wikipedia.org/wiki/Euler's_formula
|
| e (Euler's number)
| https://en.wikipedia.org/wiki/E_(mathematical_constant)
| westurner wrote:
| Is there something fundamental here - with e.g. radix base e
| - about countability and a continuum of reals, and maybe
| constructive interference?
| Dwedit wrote:
| The font I'm seeing this in makes the pi look like a lowercase N.
| colejohnson66 wrote:
| It's an uppercase pi. The lowercase one is what we're all
| familiar with. What's interesting is that the actual post has
| the lowercase one. Is it a bug in HN's capitalization fixer?
| bonzini wrote:
| Wouldn't the right question be "do the powers of pi have
| unusually big terms _towards the beginning_ of their continued
| fractions? " Because even one such term is enough to have a
| surprisingly good approximation, but many small terms at the
| beginning would create an approximation with a larger numerator
| and denominator, which may seem less remarkable than 355/113.
| thomasahle wrote:
| Pretty cool that you can determine the exact probability
| distribution for the digits of fraction expansion of a random
| real number:
|
| lim_{n->inf} Pr[k_n = k] = -log_2(1 - 1/(k+1)^2)
|
| And this was determine already in 1929! I think fraction
| expansions was all the rage back then.
| https://en.wikipedia.org/wiki/Gauss%E2%80%93Kuzmin_distribut...
| arutar wrote:
| Results like this are a lot of fun, and like many a reasonable
| number of cool results in number theory, are "surprisingly
| easy". There's a proof only using two ideas:
|
| 1) Birkhoff ergodic theorem, which states for a "nice"
| dynamical system, the probability that certain events occur can
| be described explicitly by an invariant distribution (see [1]),
| and
|
| 2) Continued fractions have an associated "nice" dynamical
| system (the Gauss map) which has an explicit probability
| distribution that is not too challenging to compute.
|
| Of course, writing this argument out takes a bit of work [2].
|
| In fact, the argument is structured in the exact same way as
| the fact that uniformly randomly chosen numbers in [0,1] are
| normal (i.e. the digit frequencies in a base-b expansion are
| all 1/b).
|
| However, proving such results about _specific_ numbers is
| notoriously hard [3]. As far as I am aware, there has not been
| a single irrational algebraic number proven to be normal.
| Normality of well-known constants like pi and e is also an open
| problem! I would not be surprised if proving distributional
| results for continued fraction expansions of pi is also very
| hard.
|
| [1]:
| https://en.wikipedia.org/wiki/Ergodic_theory#Ergodic_theorem...
|
| [2]:
| http://www.geometrie.tugraz.at/karpenkov/cf2011/cf2011s_7.pd...
|
| [3]:
| https://en.wikipedia.org/wiki/Normal_number#Properties_and_e...
| svat wrote:
| Yes! Not only that, here's something I found mind-blowing: for
| _almost all_ real numbers, the nth root of the nth convergent
| 's denominator has as limit the _same_ value, and that value is
| e^(pi^2 /12ln2).
|
| Am typing from phone so can't write it down here properly, but
| some details at an old blog post of mine:
| https://shreevatsa.wordpress.com/2010/04/30/some-incredibly-...
| madcaptenor wrote:
| there's also Khinchin's constant - for almost all real
| numbers, the geometric mean of the first n continued fraction
| coefficients approaches a constant K as ngoes to infinity. K
| is about 2.68 (the geometric mean of the Gauss-Kuzmin
| distribution).
|
| I like to pair this with a more trivial fact - for almost all
| real numbers, the arithmetic mean of the first n digits of
| the decimal expansion, as n goes to infinity, approaches 4.5.
| rich_sasha wrote:
| This is stretching my long gone education, but pi is a
| transcendental number, meaning it us not a root of any integer-
| coefficient polynomial, and these tend to not have good rational
| approximations
| [deleted]
| ykonstant wrote:
| On the contrary, due to Roth's theorem [0], transcendental
| numbers are the only ones who have a chance to be atypically
| well approximable by rational numbers.
|
| [0] https://en.wikipedia.org/wiki/Roth%27s_theorem
| [deleted]
| SilasX wrote:
| Huh? The link says Roth's Theorem is about algebraic numbers,
| which are the opposite of transcendental numbers.
|
| And I'm pretty sure 100% of integers are approximable by
| rational numbers, which has got to be at least as good a
| figure as the transcendentals can claim.
| henrydark wrote:
| Actually integers are very poorly approximable by rational
| numbers, even though each integer is very well approximated
| by a single specific rational number
|
| The definition of "well approximated" is that there are
| infinitely many good approximations, not just one really
| good one, and this is what integers, and algebraic numbers
| in general, fail to have
| SilasX wrote:
| Every integer has an infinite number of combinations of
| integers that add up to it.
| henrydark wrote:
| Indeed!
|
| Mathematics is a lot about defining something and then
| proving stuff (and sometimes going the other way around).
| Different combinations giving the same approximation are
| considered a single approximation, namely the result of
| the combination
| rich_sasha wrote:
| Ah so just off by 1 :)
|
| Now I remember, the "canonical" transcendental number is sum
| of reciprocals of n! Which is almost rational.
| ykonstant wrote:
| Precisely. In fact, that features in many proofs of
| transcendence: you find too good rational approximations,
| which contradicts things like Roth's theorem, so your
| number cannot be algebraic.
| madcaptenor wrote:
| IIRC the first number proven to be transcendental was
| Liouville's number
|
| 0.110001000000000000000001...
|
| where the digits after the decimal place are 1 in the
| n!th place and 0 otherwise. This is explicitly
| constructed to be very close to the sequence of rational
| numbers
|
| 0.1, 0.11, 0.110001, ...
|
| (I expect there's nothing special about base 10 here;
| surely the proof works in binary as well.)
| henrydark wrote:
| So actually the number you're referring to the natural base
| of logarithm, usually written "e".
|
| It turns out that e has the same irrationality measure as
| irrational algebraic numbers (2), meaning it can be
| approximated similarly well as irrational algebraic
| numbers, and not as well as some other transcendstal
| numbers like Liouville's constant
| zeroonetwothree wrote:
| Actually the irrational number that is hardest to approximate
| is phi (the golden ratio) which is not transcendental.
| hprotagonist wrote:
| "the most irrational" irrational:)
| [deleted]
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