[math-fun] Canonical sequence of rational approximants to an irrational
Given any irrational c > 0, there's a canonical sequence of rational approximants to c, defined as the maximal sequence of rationals with increasing denominators that get increasingly close to c. I calculated the first few terms for pi, e, and sqrt(2). I wasn't surprised to find the denominator sequence for pi was already in OEIS, but not those for e or sqrt(2) . . . and no written references were mentioned. Is there a name for such sequences? Is anything much known about them (e.g., speeds of approximation, growth rates of denominators, relationship to continued fractions) ? (Seems that these sequences could be used to characterize properties of the irrational, though it's not clear whether they could be any more useful than the continued fraction.) Here are the first 19 terms of each sequence: approximants to pi: 3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, 355/113, 52163/16604, 52518/16717, 52873/16830, 53228/16943, 53583/17056, ... approximants to e: 3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540, 15062/5541, 17783/6542, ... approximants to sqrt2: 1/1, 3/2, 4/3, 7/5, 17/12, 24/17, 41/29, 99/70, 140/99, 239/169, 577/408, 816/577, 1393/985, 3363/2378, 4756/3363, 8119/5741, 11482/8119, 19601/13860, 47321/33461, ... --Dan
Hi Dan and all, The sequences you describe are subsequences of the "Farey fraction" approximations. For example, see A006259 or A119015 for e. And, in turn, the denominators of the continued fraction convergents will be subsequences of your sequence, as for example A007677 for e. I would love to see your sequence in OEIS (and I hope the above gives you a little help with the cross-refs). --Joshua Zucker
Dan & others, What you've got are convergents to the continued fraction and mediants thereof. If you take the ``neg'' in place of the ``reg'', [Bill Gosper says that these suffer from malignant 2-mors, but I claim that they're b9] e.g. Pi = 4- 1/2- 1/2- 1/2- 1/2- 1/2- 1/2- 1/17- 1/294- 1/3- 1/4- ... with convergents 1/0 4/1 7/2 10/3 13/4 16/5 19/6 22/7 355/113 104348/32215 ... [E&OE -- hand calculations!] which are all > Pi. To get the ones < Pi, take the neg of -Pi -3- 1/8- 1/2- 1/2- 1/2- 1/2- 1/2- 1/2- 1/2- 1/2- 1/2- ... with convergents -1/0 3/1 25/8 47/15 68/22 91/29 113/26 135/43 157/50 179/57 201/64 223/71 245/78 267/85 289/92 311/99 333/106 355/113 ... The connexions twixt the partial quotients of the reg, the neg, and the neg of the neg, are well known to those who well know them. R. On Tue, 8 May 2007, Daniel Asimov wrote:
Given any irrational c > 0, there's a canonical sequence of rational approximants to c, defined as the maximal sequence of rationals with increasing denominators that get increasingly close to c.
I calculated the first few terms for pi, e, and sqrt(2).
I wasn't surprised to find the denominator sequence for pi was already in OEIS, but not those for e or sqrt(2) . . . and no written references were mentioned.
Is there a name for such sequences? Is anything much known about them (e.g., speeds of approximation, growth rates of denominators, relationship to continued fractions) ?
(Seems that these sequences could be used to characterize properties of the irrational, though it's not clear whether they could be any more useful than the continued fraction.)
Here are the first 19 terms of each sequence:
approximants to pi:
3/1, 13/4, 16/5, 19/6, 22/7, 179/57, 201/64, 223/71, 245/78, 267/85, 289/92, 311/99, 333/106, 355/113, 52163/16604, 52518/16717, 52873/16830, 53228/16943, 53583/17056, ...
approximants to e:
3/1, 5/2, 8/3, 11/4, 19/7, 49/18, 68/25, 87/32, 106/39, 193/71, 685/252, 878/323, 1071/394, 1264/465, 1457/536, 2721/1001, 12341/4540, 15062/5541, 17783/6542, ...
approximants to sqrt2:
1/1, 3/2, 4/3, 7/5, 17/12, 24/17, 41/29, 99/70, 140/99, 239/169, 577/408, 816/577, 1393/985, 3363/2378, 4756/3363, 8119/5741, 11482/8119, 19601/13860, 47321/33461, ...
--Dan
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participants (3)
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Daniel Asimov -
Joshua Zucker -
Richard Guy