A structural proof of the Karpelevič theorem

Brecht Verbeken; Vincent Ginis

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arXiv · 2609.26058

A structural proof of the Karpelevič theorem

Brecht Verbeken·Vincent Ginis

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Abstract

We give a self-contained proof of the Karpelevič theorem by reducing an extremal invariant polygon to a cyclic product with an exact real phase. A minimum branching count organizes the contacts, and face persistence makes a projective deformation applicable without separate tower-height cases. Convexity determines the sharp radius; explicit stochastic realizations and an independent Farey comparison identify the complete boundary. The resulting scalar equation also gives a uniform relative asymptotic for the radial deficit, including points arbitrarily close to Farey endpoints. It yields an $N^{-3}$ loss in badly approximable directions and a worst-direction loss of order $N^{-2}$ for optimal polygonal gauges.

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Brecht Verbeken, Vincent Ginis. 2026-09-23. A structural proof of the Karpelevič theorem. https://arxiv.org/abs/2609.26058

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