Search arXiv⌕ Search
Home/Archive/arXiv

arXiv · 2308.14579

Arithmetic Geometry of Non-Commutative Spaces with Large Centres

Daniel Larsson
View original source ↗Read source document ↗Export metadata ↓

Abstract

This paper introduces arithmetic geometry for polynomial identity algebras using non-commutative (formal) deformation theory. Since formal deformation theory is inherently local the arithmetic and geometric results that follow give local information that is not visible when looking at the objects from a commutative angle. For instance, it is a precise meaning to be given to two things being ``infinitesimally close'', something being obscured from view when restricting only to a commutative algebraic study. A Platonesque way of looking at this is that the commutative world is a ``shadow'' of a more inclusive non-commutative universe. The present paper aims at laying the foundation for further and deeper study of arithmetic and geometry using non-commutative geometry and non-commutative deformation theory.

Explore related subjects

math.NTmath.AGmath.RA

Keep this discovery

Explore connections, maps & timelines

BibTeXRIS

Daniel Larsson. 2023-08-28. Arithmetic Geometry of Non-Commutative Spaces with Large Centres. https://arxiv.org/abs/2308.14579

Cite the original work for its findings. Save a collection to share your selection of sources.

KEEP EXPLORING

Related papers

arXiv2026

Asymptotic density of k-almost primes

Landau's well known asymptotic formula $$N_k(x):=\ \mid\{n\leq x : Ω(n)=k\}\mid \ \sim \left( \frac{x}{\log x} \right) \frac{(\log\log x)^{k-1}}{(k - 1)!}\ \ (x \rightarrow \infty),$$ which also holds for $$π_k(x):=\ \mid\{n\leq x : ω(n)=k\}\mid,$$ is known to be fairly poor for $k > 1$, and when $k$ is allowed to tend to infinity with $x$, the study of $N_k(x)$ and $π_k(x)$ becomes very technical [1, Chapter II.6, $§$ 6.1, p.200]. I hope to show that the method described below provides not only a more accurate approach, but rather increases in its asymptotic accuracy as $k$ tends to infinity.

math.NT↗
arXiv2026

Transcendence Meets Normality: Construction of Transcendentally Normal Numbers

In this work, we study real numbers $x$ for which $p(x)$ is (absolutely) normal for every non-constant integer-valued polynomial $p$. We call such numbers transcendentally normal. We prove that almost every real number is transcendentally normal and provide an explicit construction of such a number, based on Sierpinski's covering method and novel ideas involving the so-called stretch function. In the next step, we transform this construction into an algorithm that computes the digits of a t-normal number recursively in all integer bases. Moreover, we extend our covering approach to construct and compute LIL-normal numbers whose discrepancies are of the order predicted by the law of the iterated logarithm. We also take the opportunity to discuss several interesting open problems.

math.NT↗