A computable transcendental number without eventually greedy best Egyptian underapproximations
Year: 2026
Type: preprint
Abstract: We give a deterministic rational recursion defining a specific computable transcendental number whose best Egyptian underapproximations do not form an eventually nested chain. The construction first finds, inside any rational interval in (0,1), a smaller interval on which a prescribed finite denominator prefix has no continuation that is optimal at every subsequent order. A diagonal recursion excludes all such prefixes and simultaneously avoids the zeros of every nonzero integer polynomial. Every finite stage terminates, and the resulting rational approximations have an explicit geometric error bound. The complete construction and its quantified conclusion are verified in Lean. The example lies between 1/4 and 1/3. Its best approximations use distinct positive denominators and must lie strictly below the target. All tied optimal representations are covered. This preprint supplies a specific effective construction in the setting of Erdős Problem #206; Kovač's earlier measure-zero theorem and Kovač–Tang's result for rational targets are cited in the paper. Formalization and reproduction materials: https://github.com/yuhangshi888/erdos-206-explicit-counterexample Version 1.0.0, 10 October 2026. The paper files are published with an all-rights-reserved notice. The GitHub code is separately licensed under GNU AGPL version 3 only.
Author: Shi, Yuhang
Institution: Zhengzhou University
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7221330146
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