Diophantine approximation with the constant right-hand side of inequalities on short intervals
https://doi.org/10.29235/1561-8323-2021-65-4-397-403
Abstract
In the metric theory of Diophantine approximations, one of the main problems leading to exact characteristics in the classifications of Mahler and Koksma is to estimate the Lebesgue measure of the points x ∈ B ⊂ I from the interval I such as the inequality | P (x) | < Q-w, w > n, Q >1 for the polynomials P(x) ∈ Z[x], deg P ≤ n, H(P) ≤Q is satisfied. The methods of obtaining estimates are different at different intervals of w change. In this article, at w > n +1 we get the estimate µ B< c1(n) Q – (w-1/n). The best estimate to date was c2(n) Q –(w- n/n).
About the Authors
V. I. Bernik
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Bernik Vasiliy I. – D. Sc. (Physics and Mathematics), Professor, Chief researcher
11, Surganov Str., 220072, Minsk
D. V. Vasilyev
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Vasilyev Denis V. – Ph. D. (Physics and Mathematics)
11, Surganov Str., 220072, Minsk
E. V. Zasimovich
Institute of Mathematics of the National Academy of Sciences of Belarus
Belarus
Belarus
Zasimovich Elena V. – Postgraduate student
11, Surganov Str., 220072, Minsk
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