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arXiv · 2011.10267

Isolated points of Diophantine sets

Fernando Argentieri
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Abstract

Let $γ\in(0;\frac{1}{2}),τ\geq 1$ and define the "$γ,τ$ Diophantine set" as: $$D_{γ, τ}:=\{α\in (0;1): ||qα||\geq\fracγ{q^τ}\quad\forall q\in\Bbb{N}\},\qquad ||x||:=\inf_{p\in\Bbb{Z}}|x-p|.$$ We analyze the topology of these sets and we show that generally they have isolated points.

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Fernando Argentieri. 2020-11-20. Isolated points of Diophantine sets. https://arxiv.org/abs/2011.10267

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