Open AccessDiophantine approximations with a constant right-hand side of inequalities on short intervals. 1
Author(s) -
В. И. Берник,
Natalia Budarina,
E. V. Zasimovich
Publication year - 2021
Publication title -
doklady nacionalʹnoj akademii nauk belarusi
Language(s) - English
Resource type - Journals
eISSN - 2524-2431
pISSN - 1561-8323
DOI - 10.29235/1561-8323-2021-65-5-526-532
Subject(s) - mathematics , diophantine approximation , combinatorics , diophantine equation , integer (computer science) , lebesgue measure , constant (computer programming) , lebesgue integration , upper and lower bounds , discrete mathematics , degree (music) , metric (unit) , mathematical analysis , physics , operations management , computer science , acoustics , economics , programming language
The problem of finding the Lebesgue measure of the set B 1 of the coverings of the solutions of the inequality, ⎸Px⎹ 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 B 1 <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.