Year: 2023
Type: preprint
Abstract: We examine a unique construction of the real numbers which proceeds directly from the integers using approximately linear-endomorphisms with finite error, called near-endomorphisms. In this paper, we show that the set of near-endomorphisms forms a complete ordered field isomorphic to the reals. Moreover, we show that there are uncountably many near-endomorphisms without reference to the reals. We then investigate a natural extension of near-endomorphisms, which we call quasi-homomorphisms, to other abelian groups. Extending prior results about the construction of the $p$-adic numbers and the rational adele ring, we find the ring of near-endomorphisms of certain localizations of the integers, and suggest further directions for exploration.
Source: arXiv (Cornell University)
Authors: Kumar, AJ, Long, Reese, Tung, Andrew, Wong, Ivan
Language: English
Subfield: Mathematical Physics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W4387559215