A result from diophantine approximation which has applications in economics |
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Authors: | H C Cheng A D Pollington |
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Institution: | (1) Department of Economics, University of Southern California, 90007 University Park, Los Angeles, California, USA |
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Abstract: | We say that a real number allows poor approximations if we can find 0< = ( )<1 and a sequence of integers n12<... such that for all rationals p/q with q n. we have | –.p/q| > Kn
j
–l–
where K is a constant depending only on .In this note we prove that the set of numbers which allow poor approximations are precisely the very well-approximable numbers.The existence of numbers with poor approximations has been used by Cheng 1] to show the existence of a dense set of economies whose cone converges to the Walras equilibrium as slowly as 0(n–1/2– ) after n replications. |
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