On Rational Sets in Euclidean Spaces and Spheres


IFor a positive rational ll, we define the concept of an ll-elliptic and an ll-hyperbolic rational set in a metric space. In this article we examine the existence of (i) dense and (ii) infinite ll-hyperbolic and ll-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 ll-ellitpic and ll-hyperbolic rational infinite sets in unit spheres and Euclidean spaces for certain values of ll 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 TT of the kk-dimensional unit sphere SkS^k has an antipodal pair if both x,βˆ’x∈Tx,-x\in T for some x∈Skx\in S^k. In this article, we prove that there does not exist a dense rational set TβŠ‚S2T\subset S^2 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 SkS^k is equivalent to the existence of a dense 22-hyperbolic rational set in SkS^k which is further equivalent to the existence of a dense 1-elliptic rational set in the Euclidean space Rk\mathbb{R}^k.Comment: 20 page

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