Agafonov's Proof of Agafonov's Theorem: A Modern Account and New Insights

TL;DR

This paper offers a modern, clearer account of Agafonov's proof of his theorem, providing new insights and historical context to enhance understanding of the original work.

Contribution

It recasts Agafonov's original proof using simpler methods and adds clarifications, making the theorem more accessible and providing new insights into its underlying phenomena.

Findings

A modern, simplified proof of Agafonov's theorem

Enhanced understanding of the theorem's underlying phenomena

Historical context clarifies the development of the proof

Abstract

We give a modern account of Agafonov's original proof of his eponymous theorem. The original proof was only reported in Russian in a journal not widely available, and the work most commonly cited in western literature is instead the English translation of a summary version containing no proofs, and the main proof relied heavily on material well-known in Russian mathematical circles of the day, which perhaps obscures the main thrust of argumentation for modern readers.Our present account recasts Aganofov's arguments using more basic building blocks than in the original proof, and contains some further embellishments to Agafonov's original arguments, made in the interest of clarity. We posit that the modern account provides new insight to the underlying phenomena of the theorem.We also provides some historical context to Agafonov's work, including a short description of some of the ideas that led to Agafonov's own proof, especially emphasizing the important work of Postnikova.