Simultaneous Diophantine approximation on the three dimensional Veronese curve

2025/07/29 by Badziahin, Dmitry
#11J13 #11J54 #11J82 #11K55 #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.2507.21401

Abstract

We compute the Hausdorff dimension of the set of simultaneously λ-well approximable points on the Veronese curve in \RR³ for 1/3≤ λ≤ 3/5. This range for λ was predicted in the conjecture of Beresnevich and Yang from~\citeber_yan₂023. To the best of the author's knowledge, this makes \VVV₃ the first nondegenerate curve in \RRⁿ, n≥ 3, to confirm the lower bound part of this conjecture.

Citations

• Simultaneous Diophantine approximation to points on the Veronese curve
• Rational Points Near Manifolds, Homogeneous Dynamics, and Oscillatory Integrals
• Khintchine's theorem and Diophantine approximation on manifolds
• Flows on homogeneous spaces and Diophantine approximation on manifolds

Related

• Simultaneous Diophantine approximation to points on the Veronese curve
• Diophantine approximation on curves
• Vertical shift and simultaneous Diophantine approximation on polynomial curves
• On geometry of numbers and uniform rational approximation to the Veronese curve
• Diophantine approximation on planar curves and the distribution of rational points