A note on exact approximations
Based on M. Hall's theorem we prove a simple result dealing with real numbers which admit exact approximations by rationals.
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arXiv subjects
Sergei Pitcyn
Publications and source records attributed to Sergei Pitcyn.
A note on general isolation result in Diophantine Approximation
In the present paper we give very simple general statements which deal with approximation of a real number by rationals and are related to isolation phenomenon. In particular we study functions $ f(x)>f_1(x)>0$ such that existence of solutions $\frac{p}{q}$ of Diophantine inequality $ \left| α-\frac{p}{q}\right|< \frac{f(q)}{q^2} $ leads to the existence of solutions of inequality $ \left| α-\frac{p}{q}\right|< \frac{f_1(q)}{q^2} $.
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On discrete part of Dirichlet spectrum
Recently J.Hančl obtained a result which improves on approximations to real numbers which correspond to the discrete part of Lagrange spectrum. In the present paper we prove a similar result related to the discrete part of Dirichlet spectrum.
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