Deformation Quantization and Irrational Numbers

TL;DR

This paper explores a novel form of Diophantine approximation inspired by deformation quantization, demonstrating its existence for all real numbers and analyzing special cases like rationals and quadratic irrationals.

Contribution

It introduces a new approximation framework linked to Laurent polynomials and deformation quantization, expanding the understanding of number approximation methods.

Findings

Approximation exists for all real numbers.

Special cases like rational and quadratic irrationals are analyzed.

The approach connects number theory with symplectic geometry.

Abstract

Diophantine approximation is the problem of approximating a real number by rational numbers. We propose a version of this in which the numerators are approximately related to the denominators by a Laurent polynomial. Our definition is motivated by the problem of constructing strict deformation quantizations of symplectic manifolds. We show that this type of approximation exists for any real number and also investigate what happens if the number is rational or a quadratic irrational.