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This research examines the recurrence behavior of Birkhoff sums for a zero-mean Hölder function on the circle under certain rotation conditions.

February 22, 2026Theory of Probability and Its Applications

On the Recurrence of Birkhoff Sums for a Circle Rotation and a Hölder Function

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This research examines the recurrence behavior of Birkhoff sums for a zero-mean Hölder function on the circle under certain rotation conditions.
Analyzed properties of zero-mean Hölder functions on the circle.
Investigated conditions under which irrational rotations are approximated by rational ones.
Studied the sequence of Birkhoff sums at various points on the circle.
Established that under specific approximation conditions, Birkhoff sums remain finite across the circle.
Identified that the nature of the rotation affects the finiteness of Birkhoff sums.

Implication

We explore Birkhoff sums for a zero-mean Hölder function on a circle, indicating implications for irrational rotations.

Abstract

We consider a zero-mean Hölder function on the circle. If an irrational rotation of the circle is sufficiently well approximable by rational rotations, then the sequence of values of the Birkhoff sums remains finite at each point of the circle.

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Cite This Study

A. V. Kochergin (2026) studied this question.

synapsesocial.com/papers/699a9cc6482488d673cd28fa https://doi.org/10.1137/s0040585x97t992628

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Implication

We explore Birkhoff sums for a zero-mean Hölder function on a circle, indicating implications for irrational rotations.

On the Recurrence of Birkhoff Sums for a Circle Rotation and a Hölder Function

Key Points

Implication

Abstract

Discussion

Cite This Study

On the Recurrence of Birkhoff Sums for a Circle Rotation and a Hölder Function

Key Points

Implication

Abstract

Discussion

Cite This Study

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