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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