## Presentation Type

Poster

## Start Date

5-8-2024 11:00 AM

## End Date

5-8-2024 1:00 PM

## Subjects

number theory, ergodic theory, continued fraction approximant, Fibonacci sequence, Birkhoff sum, rotation

## Advisor

Isabelle Shankar

## Student Level

Undergraduate

## Abstract

This research lies at the intersection of number theory and dynamical systems. We study the statistics of constant rotations by irrational numbers and investigate the relations between those statistics and the continued fraction approximates of a given irrational number.

This poster, which accompanies a written thesis, presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Our research is ongoing in several directions. We are investigating possible generalizations of the result for the golden mean to other numbers such as the silver mean, and other "metallic means". We are also exploring conjectures regarding the distributions of particular combinations of rotation number and size of the Birkhoff sum.

## Creative Commons License or Rights Statement

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## Persistent Identifier

https://archives.pdx.edu/ds/psu/41913

*Poster pdf*

Birkhoff Summation of Irrational Rotation: A Surprising Result for the Golden Mean

This research lies at the intersection of number theory and dynamical systems. We study the statistics of constant rotations by irrational numbers and investigate the relations between those statistics and the continued fraction approximates of a given irrational number.

This poster, which accompanies a written thesis, presents a surprising result that the difference in a certain sums of constant rotations by the golden mean approaches exactly 1/5. Specifically, we focus on the Birkhoff sums of these rotations, with the number of terms equal to squared Fibonacci numbers. The proof relies on the properties of continued fraction approximants, Vajda's identity and the explicit formula for the Fibonacci numbers.

Our research is ongoing in several directions. We are investigating possible generalizations of the result for the golden mean to other numbers such as the silver mean, and other "metallic means". We are also exploring conjectures regarding the distributions of particular combinations of rotation number and size of the Birkhoff sum.