Uniform threshold for the Hausdorff-dimension law on nondegenerate curves

Determine whether the supremal exponent up to which the Hausdorff-dimension heuristic holds uniformly for all nondegenerate curves in \(\mathbb{R}^n\) satisfies \(\tau_{n,1}=3/(2n-1)\), as conjectured by Beresnevich and Yang.

Background

The paper introduces τn,1\tau_{n,1} as the supremum of the approximation exponents up to which the expected Hausdorff-dimension formula dim⁡(C∩Wn(λ))=(2−(n−1)λ)/(1+λ)\dim(C\cap\mathcal W_n(\lambda))=(2-(n-1)\lambda)/(1+\lambda) is expected to hold uniformly for nondegenerate curves C⊂RnC\subset\mathbb{R}^n. Existing results establish this formula only in ranges near the Dirichlet exponent, while broader lower-bound results are known.

Beresnevich and Yang conjectured the uniform threshold τn,1=3/(2n−1)\tau_{n,1}=3/(2n-1). The present paper determines the dimension theory for the cubic Veronese curve V3V_3 and observes that the conjectural threshold is not a transition point for that particular curve; it does not establish or refute the conjecture for the full class of nondegenerate curves.

References

Writing \tau_{n,1} for the supremum of the exponents up to which dim_heur is expected to hold uniformly for nondegenerate curves in Rn, Beresnevich and Yang conjectured that \begin{equation}\label{eq:tau-curve-conjecture} \tau_{n,1}=\frac{3}{2n-1}. \end{equation}

eq:tau-curve-conjecture:

τn,1=32n−1.\tau_{n,1}=\frac{3}{2n-1}.

dim_heur:

dim⁡(M∩Wn(λ))=n+11+λ−(n−m).\dim\bigl(M\cap\mathcal W_n(\lambda)\bigr) = \frac{n+1}{1+\lambda}-(n-m).

— Simultaneous Diophantine approximation on the three-dimensional Veronese curve: the complete Hausdorff dimension story  (2608.25335 - Badziahin et al., 26 Aug 2026) in Introduction, equation (\ref{eq:tau-curve-conjecture})