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Approximating with Egyptian fractions

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July 18, 2022
algebraolympic revengeinequalities

Problem Statement

Determine if there exist positive real numbers x,αx, \alpha, so that for any non-empty finite set of positive integers SS, the inequality ∣x−∑s∈S1s∣>1max⁡(S)α\left|x-\sum_{s\in S}\frac{1}{s}\right|>\frac{1}{\max(S)^\alpha} holds, where max⁡(S)\max(S) is defined as the maximum element of the finite set SS.
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