A mathematical preprint reports an exact formula for the Hausdorff dimension of simultaneously λ-well-approximable points on the three-dimensional Veronese curve. Hausdorff dimension is a mathematical measure of the size of a set. For λ ≥ 1/3, the dimension of W₃(λ) ∩ V₃ is max{(2−2λ)/(1+λ), 2/[3(1+λ)]}. In plain terms, the answer is whichever of two competing expressions is larger.
The result is about mathematical sets and integer approximation vectors, not an empirical sample. The proof asks how generic points and a special family linked to tangent subspaces contribute to the dimension. The final formula compares those two analytical contributions.
The proof splits the cases
The first division is algebraic. The proof identifies special points through a zero cubic discriminant and treats generic points as the complementary class. On the generic side, it counts equivalence classes of irreducible and reducible binary cubic forms.
That counting argument produces a bound that depends on the scale being examined, rather than a single fixed total. For a compact interval I, the number of generic points near V₃ is bounded by Q²δ² up to an implied constant depending on I, provided Q ≥ 2 and 0 < δ < 1/2. Because the constant can depend on I, this is a bound tied to the interval and the chosen scales, not a fixed numerical total.
The second contribution
The special points require a separate construction. The proof builds explicit special approximants, uses a Cantor-type construction and the mass distribution principle to obtain the lower bound, then uses an explicit lattice parametrization for the matching upper bound. Theorem 3 gives the special-point contribution as max{(2−2λ)/(1+λ), 2/[3(1+λ)]} over the λ range stated there.
The comparison between the two terms changes at λ = 2/3. The paper describes this as a phase transition, with tangent-subspace arithmetic dominant beyond that point. For generic points, Corollary 1 gives an upper bound of max{(2−2λ)/(1+λ), 0} over its stated λ range. That generic expression is an upper limit, not an equality throughout every part of the range.
What the theorem claims
The authors say the theorem fills a previously open interval, 3/5 < λ < 7/9, and that 3/5 is not a transition point for this curve. They also qualify their novelty assessment, saying that, to their knowledge, the work gives the first complete dimension law for a nondegenerate, non-planar curve.
Using the exponent formulation, the paper says Theorem 1 determines the complete upper-level dimension function for λ₃. It also answers positively, for n = 3, whether the spectrum is the whole interval [1/n, ∞]. These statements are made within the three-dimensional setting of the result.
A result with a defined scope
The paper is a mathematical preprint, identified in its front matter as arXiv:2608.25335v1 in math.NT and dated 26 Aug 2026. It analyzes mathematical sets, curves and integer approximation vectors rather than an empirical participant sample.
The theorem is stated for the three-dimensional Veronese curve, and the authors’ novelty statement is qualified as a claim made to their knowledge. No funding or conflict-of-interest statement is reported.
Paper data and sources
Original title: Simultaneous Diophantine approximation on the three-dimensional Veronese curve: the complete Hausdorff dimension story
Authors: Dmitry Badziahin, Nikita Shulga
Journal/Repository: arXiv
Status: Preprint, not yet peer-reviewed
First online: 2026-08-26
DOI: Not available
Original paper · Full text