Adams80a W. W. Adams. The best two-dimensional diophantine approximation constant for cubic irrationals. Pacific J. Math., 91:29--30, 1980. Home/Search Document Not in Database Summary Related Articles Check |
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On the Furtwaengler algorithm for simultaneous rational.. - Briggs (Correct)
.... ; p 2 ) 2 Z 2 ; q 2 Z; 1 ; 2 ) 2 R 2 n Q 2 , let c( q) qkq k 2 ; c( lim inf q 1 c( q) The two dimensional sup norm simultaneous diophantine approximation constant is then c 2 = sup c( Despite much work over the last few decades [Cassels 1955, Adams 1969b, Adams 1980, Cusick 1974b, Cusick 1983, Szekeres 1984, Szekeres 1985, Kratz 3 1999] the value of c 2 is unknown, though folk lore suggests that its value is 2=7. Adams 1969b] has shown that this is the correct value if we restrict the pair ( 1 ; 2 ) to cubic number elds (see below) but his result ....
....formed by any three successive best approximation triples is always unity. I have not been able to nd any other pair showing such regularity. I will discuss this eld further in section 7. It is known that cubic number elds play a special role here [Adams 1969b, Adams 1969a, Adams 1971, Adams 1980, Cusick 1974b, Cusick 1977] and that pairs such as the above produce some kind of approximate periodicity. Figure 1 shows the function c( q) at best approximants for the elds of discriminant 23; 31; 44; 49; 81 with the integral bases given in Table 1. I investigated periodicity by looking ....
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Adams, W. W. [1980], `The best two-dimensional diophantine approximation constant for cubic irrationals', Pacic J. Math, 91, 29-30.
Numbers Approximating Badly - Keith Briggs Department (Correct)
.... C(ff; p; q) q max(jqff 1 Gamma p 1 j 2 ; jqff 2 Gamma p 2 j 2 ) and C(ff) lim inffC(ff; p; q) p 2 Z 2 ; q 2 Zg: 1 The two dimensional (max norm) simultaneous diophantine approximation constant is then C 2 = supfC(ff) ff 2 R 2 g: Despite much work over the last few decades [1, 2, 3, 4, 5, 6, 7], the value of C 2 is unknown, though folk lore suggests that its value is 2=7. Adams [2] has shown that this is the correct value if we restrict the pair (ff 1 ; ff 2 ) to cubic number fields (see below) but his result does not give us a constructive procedure to identify pairs with large C 2 ....
W. W. Adams. The best two-dimensional diophantine approximation constant for cubic irrationals. Pacific J. Math,, 91:29--30, 1980.
Version of Wed 2001 Mar 21 16:53 - Szekeres Szekeres The (Correct)
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Adams80a W. W. Adams. The best two-dimensional diophantine approximation constant for cubic irrationals. Pacific J. Math., 91:29--30, 1980.
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