Open AccessCounting algebraic numbers in short intervals with rational points
Author(s) -
В. И. Берник,
Friedrich Götze,
Николай Иванович Калоша
Publication year - 2019
Publication title -
žurnal belorusskogo gosudarstvennogo universiteta. matematika, informatika/žurnal belorusskogo gosudarstvennogo universiteta. matematika, informatika
Language(s) - English
Resource type - Journals
eISSN - 2617-3956
pISSN - 2520-6508
DOI - 10.33581/2520-6508-2019-1-4-11
Subject(s) - algebraic number , mathematics , conjecture , interval (graph theory) , real number , algebraic extension , distribution (mathematics) , discrete mathematics , rational number , combinatorics , algebraic cycle , mathematical analysis , differential algebraic equation , ordinary differential equation , differential equation
In 2012 it was proved that real algebraic numbers follow a nonuniform but regular distribution, where the respective definitions go back to H. Weyl (1916) and A. Baker and W. Schmidt (1970). The largest deviations from the uniform distribution occur in neighborhoods of rational numbers with small denominators. In this article the authors are first to specify a gene ral condition that guarantees the presence of a large quantity of real algebraic numbers in a small interval. Under this condition, the distribution of real algebraic numbers attains even stronger regularity properties, indicating that there is a chance of proving Wirsing’s conjecture on approximation of real numbers by algebraic numbers and algebraic integers.