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Fractions in (0,1) with bounded denominators

Source: Tuymaada 2023 Senior P8

July 7, 2023

Problem Statement

Given is a positive integer nn. Let AA be the set of points x∈(0;1)x \in (0;1) such that ∣x−pq∣>1n3|x-\frac{p} {q}|>\frac{1}{n^3} for each rational fraction pq\frac{p} {q} with denominator q≤n2q \leq n^2. Prove that AA is a union of intervals with total length not exceeding 100n\frac{100}{n}.
Proposed by Fedor Petrov
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