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rational NT involving floor function

Source: Yugoslav TST 1974 P1

May 29, 2021
floor functionRationalsnumber theoryalgebrafunction

Problem Statement

Assume that aa is a given irrational number.
(a) Prove that for each positive real number ϵ\epsilon there exists at least one integer q≥0q\ge0 such that aq−⌊aq⌋<ϵaq-\lfloor aq\rfloor<\epsilon. (b) Prove that for given ϵ>0\epsilon>0 there exist infinitely many rational numbers pq\frac pq such that q>0q>0 and ∣a−pq∣<ϵq\left|a-\frac pq\right|<\frac\epsilon q.
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