A note on general isolation result in Diophantine Approximation

2025/09/30 by Pitcyn, Sergei, Moshchevitin, Nikolay
#11J04 #11J06 #11J13 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2509.25628

Abstract

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)>f₁(x)>0 such that existence of solutions (p)/(q) of Diophantine inequality | α-(p)/(q)|< (f(q))/(q²) leads to the existence of solutions of inequality | α-(p)/(q)|< (f₁(q))/(q²) .

Citations

• A quantitative framework for sets of exact approximation order by rational numbers
• Difference of irrationality measure functions
• Approximations to two real numbers

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• Diophantine approximation with prime denominator in real quadratic function fields
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• Exponents of Diophantine Approximation in dimension two