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E(x)={[nx]: n\in N}, a> 1, b>0,E ( b) \subset E(a),. then b/a in N

Source: Indian Postal Coaching 2008 set 1 p1

May 25, 2020
floor functionnumber theoryFractionnaturalalgebrairrational

Problem Statement

For each positive x∈R x \in \mathbb{R}, define
E(x)={[nx]:n∈N} E(x)=\{[nx]: n\in \mathbb{N}\}
Find all irrational α>1 \alpha >1 with the following property: If a positive real β \beta satisfies E(β)⊂E(α) E(\beta) \subset E(\alpha). then βα \frac{\beta}{\alpha} is a natural number.
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