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.

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May 8th, 11:00 AM May 8th, 1:00 PM

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.