Approximation of π by rationals of the form a/b2 .

Oleg Karpenkov conjectured that any real number ξ can be approximated by a rational of the form a/b2 such that the inequality

| ξ - a/b2| < c/b3,

has infinitely many solutions in integers a and b [1] (see also Hardy and Littlewood [3], Heilbronn [4] and Zaharescu [5]).
The table below shows the values of b corresponding to approximations of π=3.14159 ... that are less in absolute value than 1/b3. The value of a can be computed as the nearest integer to πb2. The table was computed by means of an algorithm [2] that performs b1/2 polynomial time operations in average to compute the items in it.
A PARI script to calculate the convergents of the form a/b2 for any real can be found here.
#b #b #b
01
12 36207207 71765799023035418
233726528072924764108790473
3638537766731063349625535625
4739925036741180700047859443
58 401693817751392877612852012
613 411803181762265120792743544
728422053555775655090457146592
83243115185267810791046500563708
94144667975477929523781971207577
1050 45903289408029610072444366717
118146994093718171502903187242438
12854710534541582199860004524767978
13113 48201049505 83227906317501081783
141414934359479584399720009049535956
151985065407588985438348720358029255
162675193497932386526976478335568535
1765952106136887487530420552026510882
181014531000748651788599580013574303934
1914455410143034860 89775069693146270992
201650551709886612690809695991456163835
2117385630663791044912040782552597732857
222028 57242095640151922256217400888336760
232163 58817685025481932603652578578390773
242658 591100586602840 943397660065732068041
253091601223837592174 9514321981424900356103
2631446110033284088464
2710224 6229365202693872
28132716335649558528135
29134516445779483485139
30187226555318927739742
31238716673713181052162
322690267101266519548801
334035368103747516485092
3410052569254245202820313
35180850 70326784622391234

[1]  Oleg Karpenkov "Approximating reals by rationals of the form a/b2"

[2]  I. Jimenez Calvo, "An algorithm to approximate reals by rationals of the form a/b2".

[3]  G. H. Hardy and J. E. Littlewood. "Some problems of diophantine approximation I: The fractional part of nk&theta". Acta Math. 37, 155-191 (1914).

[4]  H. Heilbronn. "On the distribution of the sequence n2&theta (mod 1)". Quart. J. Math. Oxford Ser. 19, 249-256 (1948).

[5]  A. Zaharescu. "Small values of n2&alpha". Invent. Math. 121, 379-388, (1995).



Last updated: April, 13, 2007

 Homepage of Ismael Jiménez Calvo