IFor 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-ellitpic
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-ellitpic and l-hyperbolic rational infinite sets in unit
spheres and Euclidean spaces for certain 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,βxβT for some xβSk. In this article,
we prove that there does not exist a dense rational set TβS2 which
has an antipodal pair by assuming Bombieri-Lang Conjecture for surfaces of
general type. 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 Sk which is further equivalent to the existence of a dense 1-elliptic
rational set in the Euclidean space Rk.Comment: 20 page