Simultaneous Diophantine approximation on the three dimensional Veronese curve

Dmitry Badziahin

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arXiv · 2507.21401

Simultaneous Diophantine approximation on the three dimensional Veronese curve

Dmitry Badziahin

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Abstract

We compute the Hausdorff dimension of the set of simultaneously $\lambda$-well approximable points on the Veronese curve in $\RR^3$ for $1/3\le \lambda\le 3/5$. This range for $\lambda$ was predicted in the conjecture of Beresnevich and Yang from~\cite{ber_yan_2023}. To the best of the author's knowledge, this makes $\VVV_3$ the first nondegenerate curve in $\RR^n$, $n\ge 3$, to confirm the lower bound part of this conjecture.

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Dmitry Badziahin. 2025-07-29. Simultaneous Diophantine approximation on the three dimensional Veronese curve. https://arxiv.org/abs/2507.21401

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