Abstract—Emptiness is usually considered to be absolute and equal to null or zero. This paper proposes to analyze conditions close to emptiness. A relaxation of abstract concepts of zero and emptiness is used to analyze sparsely populated sets. The proposed definition of emptiness is based on the Riemann’s zeta function and depends on the perspective of the point from which emptiness is perceived. The new symmetric and asymmetric analytic continuations are proposed to resolve the asymptotic behavior of the zeta function near the pole. Different levels of infinity and emptiness are associated with “analytic convergence” using the asymmetric analytic continuation for the zeta function. The classifications of emptiness and infinity improve efficiency of data search in sparsely populated large sets, especially, when the data retrieval effort depends on the distance to remote data source.

Index Terms—Sparse, population, infinity, sets, zero, emptiness, classification, zeta function, deformation, search.

Kiamran Radjabli is with Utilicast, La Jolla, California, USA (email: kradjabli@utilicast.com).

Cite: Kiamran Radjabli, "Analysis of Sparsely Populated Sets," International Journal of Applied Physics and Mathematics vol. 7, no. 1, pp. 12-23, 2017.