[HN Gopher] Maybe powers of p don't have unexpectedly good appro... ___________________________________________________________________ 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] ___________________________________________________________________ (page generated 2022-07-13 23:01 UTC)