On a problem in simultaneous Diophantine approximation: Schmidt’s conjecture
By Dzmitry Badziahin, A. Pollington, Sanju Velani. 2010
74 citations recorded by the source corpus.
For any i;j > 0 with i + j = 1, let Bad(i;j) denote the set of points (x;y) 2 R 2 for which maxfkqxk 1=i ; kqyk 1=j g > c=q for all q 2 N. Here c = c(x;y) is a positive constant. Our main result implies that any nite intersection of such sets has full dimension. This settles a conjecture of Wolfgang M. Schmidt in the theory of simultaneous Diophantine approximation.
Subfields: Analytical Chemistry, Mathematical Physics, Statistical and Nonlinear Physics
View Paper: https://www.semanticscholar.org/paper/83f6e6ce6ad87a698be80651020581c73bf9cfb2
How researchers see this paper
The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.
In Analytical Chemistry
In Mathematical Physics
- Victor Beresnevich
- Felipe A. Ram'irez
- Sanju Velani
- Jinpeng An
- Yann Bugeaud
- Bao-Wei Wang
- Dmitry Kleinbock
- Jun Wu
In Statistical and Nonlinear Physics
Citations in the corpus
- Badly approximable points on manifolds (2013, background): “ntil recently there has been no success in relation to Davenport’s problem even for planar curves, let alone manifolds in higher dimension. The aforementioned work of Badziahin, Pollington and Velani [BPV11] was the first step forward. Very recently, assuming (5), Badziahin and Velani [BV14] have pro”
- Two-dimensional badly approximable vectors and Schmidt's game (2012, background): “On the other hand, the main theorem in [2] implies that if (F+ n ) ∞ n=1 is a sequence of rays in D + satisfying a certain technical assumption, then the set {h ∈ H : F n hΓ is bounded, ∀n ≥ 1} (1.”
- Metric Diophantine Approximation : aspects of recent work (2016, background): “Schmidt’s Conjecture was proved in [6] by establishing the following stronger statement regarding the intersection of Bad(i, j) sets with vertical lines Lα := {(α, y) : y ∈ R} ⊂ R2.”