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Journal of the Ramanujan Mathematical Society

Volume 38, Issue 2, June 2023  pp. 107–120.

On Rational Sets in Euclidean Spaces and Spheres

Authors:  C. P. Anil Kumar
Author institution: School of Mathematics, Harish-Chandra Research Institute, HBNI, Chhatnag Road, Jhunsi, Prayagraj, Allahabad 211 019, India.

Summary:  For a positive rational l, we define the concept of an l-elliptic and an l-hyperbolic rational set in a metric space. In this article we examine the existence of (i) dense and (ii) infinite l-hyperbolic and l-elliptic rationals subsets of the real line and unit circle. For the case of a circle, we prove that the existence of such sets depends on the positivity of ranks of certain associated elliptic curves. We also determine the closures of such sets which are maximal in case they are not dense. In higher dimensions, we show the existence of l-elliptic and l-hyperbolic rational infinite sets, in general position, on unit spheres and in Euclidean spaces, for those values of l which satisfy a weaker condition regarding the existence of elements of order more than two, than the positivity of the ranks of the same associated elliptic curves. We also determine their closures. A subset T of the k-dimensional unit sphere Sk has an antipodal pair if both x, −x ∈ T for some x ∈ S{k}. In this article, we prove that there does not exist a dense rational set T ⊂ S{2} which has an antipodal pair by assuming Bombieri-Lang Conjecture for surfaces of general type. For any k ∈ N, we actually show that the existence of such a dense rational set in Sk is equivalent to the existence of a dense 2-hyperbolic rational set in S{k} which is further equivalent to the existence of a dense 1-elliptic rational set in the Euclidean space R{k}.

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