On the sharpness of a three circles theorem for discrete harmonic functions

TL;DR

This paper identifies the precise error term in an $L^2$ three circles theorem for discrete harmonic functions on ^2, using advanced interpolation and recursive techniques to overcome combinatorial challenges.

Contribution

It determines the sharp error term in an $L^2$ three circles theorem for discrete harmonic functions, advancing understanding of their growth properties.

Findings

Established the exact error term in the theorem.

Developed novel recursive and interpolation methods.

Addressed combinatorial obstacles in the proof.

Abstract

Any three circles theorem for discrete harmonic functions must contain an inherent error term. In this paper we find the sharp error term in an $L^2$-three circles theorem for harmonic functions defined in $\Zb^2$. The proof is highly indirect due to combinatorial obstacles and cancellations phenomena. We exploit Newton interpolation methods and recursive arguments.