Abstract
where u1, U2 U3 I U4 are four integral unimodular forms in x1, X2 I X3 X4X the Pi, P2, P3, P4 are four real numbers whose product is A, and ni , n2 , n3 , n4 are integers. The proof will be carried out in two stages, the results of which are stated separately in the following two theorems A and B. THEOREM A. Let a 4-dimensional space S be given, with Cartesian coordinates yl X y2 X Y3 , Y and origin 0. Let L be a set of points in S with the following properties: (i) L does not contain 0, (ii) L contains only a finite number of points in any bounded region of S, (iii) Given any one A of the coordinate axes and any positive number (, L contains a point whose distance from A is less than E (and may be zero). Then there exist four positive real numbers 'X , ,2 X3 X4 such that the ellipsoid
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