Diophantine approximations with a constant right-hand side of inequalities on short intervals. 1
https://doi.org/10.29235/1561-8323-2021-65-5-526-532
Abstract
The problem of finding the Lebesgue measure 𝛍 of the set B1 of the coverings of the solutions of the inequality, ⎸Px⎹ <Q−w, w>n , Q ∈ N and Q >1, in integer polynomials P (x) of degree, which doesn’t exceed n and the height H (P) ≤ Q , is one of the main problems in the metric theory of the Diophantine approximation. We have obtained a new bound 𝛍B1 <c(n)Q−w+n, n<w<n+1, that is the most powerful to date. Even an ineffective version of this bound allowed V. G. Sprindzuk to solve Mahler’s famous problem.
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), Pro fessor, Chief researcher
11, Surganov Str., 220072, Minsk
N. V. Budarina
Institute of Technology of Dundalk
Ireland
Ireland
Budarina Nataliya V. – D. Sc. (Physics and Mathematics)
A91 K584, Dublin Road, Dundalk
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
References
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