The Norm of Gaussian Periods

TL;DR

This paper studies the asymptotic behavior of the absolute norm of Gaussian periods, providing convergence rates and connecting number theory, algebraic geometry, and diophantine approximation techniques.

Contribution

It advances understanding of Gaussian periods by establishing their norm's asymptotics and convergence rates, extending results related to Myerson's Conjecture for odd periods.

Findings

Derived asymptotic formulas for Gaussian period norms

Provided convergence rate in Myerson's Conjecture case

Connected number theory with algebraic geometry and diophantine approximation

Abstract

Gaussian periods are cyclotomic integers with a long history in number theory and connections to problems in combinatorics. We investigate the asymptotic behavior of the absolute norm of a Gaussian period and provide a rate of convergence in a case of Myerson's Conjecture for periods of arbitrary odd length. Our method involves a result of Bombieri, Masser, and Zannier on unlikely intersections in the algebraic torus as well as work of the author on the diophantine approximations to a set definable in an o-minimal structure. In the appendix we make a result of Lawton on Mahler measures quantitative.