| Title |
Small differences between cubic and rational numbers |
| Authors |
Dubickas, Artūras |
| DOI |
10.1017/S0017089526100998 |
| Full Text |
|
| Is Part of |
Glasgow mathematical journal.. Cambridge : Cambridge University Press. 2026, Early Access, p. [1-8].. ISSN 0017-0895. eISSN 1469-509X |
| Keywords [eng] |
approximation to algebraic numbers ; continued fraction ; cubic number ; irreducibility |
| Abstract [eng] |
In this paper,we showthat for each sufficiently large integerH, there is a real cubic number α and a rational number r, both of height smaller than H, such that 0<α −r <272/H4. The exponent 4 of H in this inequality is best possible. The numbers α and r are both constructed explicitly.We also show that a necessary and sufficient condition on (u, v) under which there is a positive constant c1 =c1(d, u, v) such that the inequality |α −β| ≥ c1H(α)−uH(β)−v holds for all real algebraic numbers α of degree d ≥ 2 and all real algebraic numbers β of degree d −1 is u ≥ d −1 and v ≥ d. |
| Published |
Cambridge : Cambridge University Press |
| Type |
Journal article |
| Language |
English |
| Publication date |
2026 |
| CC license |
|
