On Rational Sets in Euclidean Spaces and Spheres

TL;DR

This paper investigates the existence and properties of rational sets with specific geometric and algebraic conditions in Euclidean spaces and spheres, linking their existence to elliptic curve ranks and conjectures in number theory.

Contribution

It introduces the concepts of $l$-elliptic and $l$-hyperbolic rational sets, analyzes their existence in various spaces, and connects their properties to elliptic curve ranks and the Bombieri-Lang conjecture.

Findings

Existence of dense and infinite $l$-hyperbolic and $l$-elliptic rational sets depends on elliptic curve ranks.

No dense rational set with an antipodal pair exists in $S^2$ assuming Bombieri-Lang conjecture.

Existence of dense rational sets with certain properties is equivalent across different spaces and set types.

Abstract

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 $S^k$ has an antipodal pair if both $x,-x\in T$ for some $x\in S^k$. In this article, we prove that there does not exist a dense rational set $T\subset S^2$ 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 $S^k$ 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 $\mathbb{R}^k$.