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Nearly parallel vectors

Published online by Cambridge University Press:  26 February 2010

Harold G. Diamond  and
Carl Pomerance
Harold G. Diamond
Affiliation:
Department of Mathematics, University of IllinoisUrbana, Illinois 61801.
Carl Pomerance
Affiliation:
Department of Mathematics, University of GeorgiaAthens, Georgia 30602;, University of Illinois, Urbana, Illinois 61801.
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Extract

The first problem in diophantine approximation is for a given real number ξ, and positive number x to find a fraction s/t with t ≤ x which is close to ξ. This problem can be rephrased in geometric terms. Given a vector v in ℝ2, find a vector a = (a1, a2) with integer coordinates and 1 1 ≤ al ≥ x such that the vectors a and v are nearly parallel. Simultaneous approximation of d − 1 real numbers can be recast in terms of approximation of the angle between two vectors in d-dimensional Euclidean space.

Type
Research Article
Information
Mathematika , Volume 26 , Issue 2 , December 1979 , pp. 258 - 268
Copyright
Copyright © University College London 1979

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References

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