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On the refined shrinking target property of rotations

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arxiv 1201.4568 v2 pith:GYLHNK64 submitted 2012-01-22 math.NT math.DS

classification math.NTmath.DS
keywords varphieveryincreasingirrationalmonotonepropertyrotationsshrinking
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abstract

We discuss the shrinking target property of irrational rotations. We obtain the condition of an irrational $\theta$ and monotone increasing $\varphi(n)$ such that $$ \liminf_{n \to \infty} n \varphi (n) \| n\theta - s \| = 0 \text{ for almost every } s. $$ We also consider the class of irrationals for which the limit inferior is 0 for every monotone increasing $\varphi(n)$ such that $\sum_n 1/(n\varphi(n))$ diverges.

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  1. Self-Referential Leading Digits of Exponential Sequences: Arithmetic Structure and Certified Search

    math.NT 2026-07 accept novelty 7.0 of 10

    Self-prefix hits of c^m admit exact discrepancy identities, Lambert two-gap candidates, resonance rigidity, and certified O(N^{1-1/ν} polylog N) search; infinitude of 2^m starting with m remains open.

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