REVIEW 3 major objections 4 minor 20 references
Simultaneous Diophantine approximation on the three dimensional Veronese curve
T0 review · 3 major / 4 minor · reviewed 2026-08-06 · deepseek-v4-flash
Pith's one-line read For every λ between 1/3 and 3/5, the Hausdorff dimension of simultaneously λ-well approximable points on the three-dimensional Veronese curve equals (2−2λ)/(1+λ).
desk verdict A real step to the conjectured endpoint 3/5 for the Veronese curve in R^3, with the proof's weight resting on counting estimates imported from the author's earlier paper—worth a serious referee. read the letter →
The pith
A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.
The reading
What carries the argument
The central counting device is Theorem 2, which bounds the number of cubic polynomials with height at most $H$, discriminant $0 < |D(P)| \le D$, and root separation at least $R^{-1}$ by $\ll \min\{H^{2/3+\epsilon}D^{5/6},\log H \cdot D R\} + H^\epsilon D$. The count is assembled from equivalence classes of cubics under fractional linear transformations, using the number of classes with a given discriminant as an input. The surrounding proof uses dyadic blocks, the boxes $\Delta_m$ defined by closeness to tangent segments of the curve, the successive minima of these boxes, and parameters $\delta,\eta,\kappa$ that measure how far a box is from the generic case. For zero discriminant the polynomial takes the form $(ax-b)^2(cx-d)$, and the remaining work is a partition by which linear factor's root lies in the associated interval $J$.
What would settle it
For $\lambda=3/5$ and a large dyadic scale $Q=2^k$, enumerate the primitive integer vectors $q$ inside one of the boxes $\Delta_m$ used in the proof and compare their number with the right-hand side of estimate (10); a single box that exceeds that bound would invalidate the exceptional-box count and with it the upper bound behind Theorem 1.
Extended reading notes
Core claim
The central claim is Theorem 1: for $V_3 = \{(x, x^2, x^3) : x \in I\}$ with $I$ a closed interval not containing zero, and for every $1/3 \le \lambda \le 3/5$, $\dim S_3(I,\lambda) = (2-2\lambda)/(1+\lambda)$, where $S_3(I,\lambda)$ is the restriction to $V_3$ of the set of $x \in \mathbb{R}^3$ satisfying $\|qx - p\|_\infty < q^{-\lambda}$ for infinitely many integer pairs $(q,p)$. The proof supplies the upper bound needed for the formula; the matching lower bound for this range is already available from the theory of rational points near manifolds, so the equality follows. Grouping rational vectors into dyadic blocks in which the denominator $q$ lies between consecutive powers of two, the argument partitions each block by how close the rational points are to tangent segments of the curve. Each remaining case is then controlled by counting the associated cubic polynomials $P_a(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3$ whose height is bounded in terms of the block parameters. Polynomials with nonzero discriminant are counted by a new estimate for cubic polynomials with bounded height, bounded discriminant, and large root separation; polynomials with zero discriminant factor as $(ax-b)^2(cx-d)$ and are handled by a case analysis according to which root lies in the relevant approximation interval. The paper states that this makes $V_3$ the first nondegenerate curve in $\mathbb{R}^n$, $n \ge 3$, to confirm the lower-bound half of the conjecture in this range.
Load-bearing premise
The proof leans on two counting estimates imported from the author's earlier work: a height bound on the polynomial attached to each rational vector and a bound on how many rational vectors can share one small interval; the paper does not re-derive these estimates.
Editorial extensions
If this is right
- For all $\lambda$ in $[1/3,3/5]$, the Hausdorff dimension on $V_3$ is exactly $(2-2\lambda)/(1+\lambda)$, falling from $1$ at $\lambda=1/3$ to $1/2$ at $\lambda=3/5$.
- The known range for $V_3$ is extended from $\lambda \le 1/2$ to the full interval predicted for the conjecture, so the conjectured threshold $3/(2n-1)$ is attained for $n=3$ on this curve.
- The zero-discriminant analysis shows that the only delicate configurations are intervals containing rationals with denominator at most $H^{1/3}$; all other exceptional boxes contribute less than the dimension bound.
- Because the lower bound was already known, the equality on $V_3$ turns the conjecture's predicted value into a theorem for this test curve.
Reading between the lines
- The polynomial-counting estimate used here is not tied to the Veronese parametrization; a similar count should apply to any nondegenerate curve in $\mathbb{R}^3$ whose local approximating polynomials are cubic, which would widen the verified range beyond $V_3$.
- The case analysis isolates the barrier at $\lambda=3/5$: intervals that contain very simple rationals (denominator $\ll H^{1/3}$) are the ones that just barely fit inside the dimension bound. A proof for $\lambda>3/5$ would need a new way to handle exactly those intervals.
- A numerical experiment at $\lambda=3/5$ could probe whether the threshold is sharp: the proof predicts that the number of rational vectors in each exceptional box grows like $Q^{2-2\lambda+o(1)}$, and a measurable excess on dyadic scales would suggest the conjectured range is not optimal.
Signed reviews
Editorial analysis
A structured set of objections, weighed in public.
Referee Report
Summary. The paper proves that for the Veronese curve V_3 = {(x,x^2,x^3)} in R^3, the Hausdorff dimension of the set S_3(I,λ) of simultaneously λ-well approximable points equals (2−2λ)/(1+λ) for every λ ∈ [1/3, 3/5]. This matches the range predicted by the Beresnevich–Yang conjecture for nondegenerate curves. The proof is an upper-bound argument that partitions the relevant rational vectors into a hierarchy of cases: nonzero-discriminant polynomials are treated via a new counting result for cubic polynomials (Theorem 2), and zero-discriminant cases are split into several geometric subcases controlled by the parameters δ, η, κ, σ, δ*. The paper is a continuation of the author's earlier work [1], and several central counting estimates are imported from that paper.
Significance. If the proof is correct, Theorem 1 provides the first verification of the Beresnevich–Yang dimension formula for a nondegenerate curve in R^n, n ≥ 3, up to the conjectured threshold 3/(2n−1), with n = 3. The paper also contributes a new counting result for cubic integer polynomials with small discriminant and large root spread (Theorem 2), which is a substantial technical tool. The exposition is structured and the case analysis is detailed; the exponent bookkeeping in Sections 5–8 is internally coherent once the imported estimates (9)–(10) are granted. However, the main theorem is conditional on those imported estimates, and the lower-bound half of the equality is not explicitly established in this paper.
major comments (3)
- [§3, Eqs. (9)–(10)] The two counting estimates (9) and (10), imported from [1] without proof, are load-bearing for every upper-bound branch: they control S3 in §5, S4–S6 in §6, S7–S9 in §7, and S10–S11 in §8. In particular, the exponent (3−5λ)/2 in (10) appears in every subsequent estimate, and a multiplicative Q^c loss in that formula would destroy the final bound at λ = 3/5, where the target exponent (2−2λ)/(1+λ) = 1/2 is met exactly. The paper does not state the hypotheses under which (9) and (10) are valid, nor the uniformity of the implicit constants in ε, δ, and η. Please reproduce the proofs of these estimates in an appendix, or state them as formal lemmas with complete hypotheses and precise references to the corresponding results in [1].
- [§5, Proposition 1] The proof of Proposition 1 contains the unproved assertion that the number of pairs (a,b) with |a| ≤ A and |y1 + b/a| ∈ [|a|^{-t−ε}, |a|^{-t}] is bounded by ≪ A^{2−t}. This estimate is used directly to obtain (16), which in turn yields Theorem 2 and the D3 estimate. The proposition also relies on [1, Lemma 9] for the root-separation property |x_i − x_j| ≫ 1 of the reduced form R_a, and that separation lemma is not stated. These are load-bearing gaps in the new part of the proof; please supply the missing counting argument and a precise statement of the imported separation lemma.
- [Theorem 1 and §3–§8] The proof as written establishes upper bounds for the Hausdorff dimension of the sets S_j; it never explicitly proves the matching lower bound dim S_3(I,λ) ≥ (2−2λ)/(1+λ). If the lower bound for the full range 1/3 ≤ λ ≤ 3/5 is available from [1] or from Beresnevich's theorem, it should be stated explicitly and its hypotheses checked. If it is not known for λ > 1/2, then the equality in Theorem 1 is not justified. Please add a clear statement of the lower-bound theorem used and explain how it covers the range [1/3, 3/5].
minor comments (4)
- [§6, after Eq. (20)] The displayed inequality "H ≍ |P'''(x)| ≪ Q^{−(1+λ)/2 + 3η}" appears to have a sign error: equation (12) with i = n = 3 gives the exponent +(1+λ)/2 + 3η, and it is that positive exponent which yields the stated bound η ≳ −(1+λ)/6. Please correct the displayed sign.
- [§5, Theorem 2 proof] In the summation over discriminants, the intermediate expression "min{H^{2/3+ε} d^{−1/6}, log H · R^{1+ε}}" should have R, not R^{1+ε}, if it is quoting Proposition 1; the final bound in (19) is correct, but the intermediate line should be adjusted for consistency.
- [§1, Introduction] The sentence "If M is one dimensional ... the above inequality is achieved for 1/n ≤ λ ≤ 3/(2n−1)" is ambiguous: it is not clear whether this means the lower bound is known to hold throughout that range or that equality is known. Since the conjecture is still open, please clarify that the lower bound is the known part and the present paper supplies the matching upper bound.
- [Abstract] The abstract says the result "confirm[s] the lower bound part of this conjecture," but the proof establishes an upper bound. If the lower bound is already known, it would be more accurate to say the paper confirms the full dimension formula for V3 in this range; please rephrase to avoid confusion about what is new.
Circularity Check
No construction-level circularity: the dimension formula is neither a fitted input nor a definitional identity, though the proof leans heavily on previously published counting estimates from the author's earlier paper.
full rationale
The paper's derivation is not circular at the construction level. The target quantity dim S3(I, λ) is defined by a limsup of rational approximations to the Veronese curve, and the upper-bound strategy is a standard Hausdorff-Cantelli and covering argument. No parameter is fitted to the target dimension, and the formula (2−2λ)/(1+λ) is not encoded into the definitions of D(k) or the auxiliary boxes. The two key imported estimates, (9) and (10), are taken from the author's earlier paper [1] and control, respectively, the height of the associated polynomial and the number of rational vectors sharing an interval J; these are auxiliary counting statements whose assumptions do not already contain the conclusion of Theorem 1. All later exponent calculations are bookkeeping consequences of these estimates, so a failure of (9) or (10) would be a correctness risk, but reliance on a prior published paper is not circular reasoning. The same holds for the invocations of [1, Lemma 6], [1, Lemma 9], and [1, Proposition 5]: they are used as established facts rather than re-proved, but they do not reduce the theorem to its own assumptions. There is no fitted-input-called-prediction step, no self-definitional identity, and no renaming of a known empirical pattern as a new unification. The score of 2 reflects the heavy self-citation footprint and the fact that the proof is not self-contained, while the central claim remains independent of the paper's own construction.
Assumptions & free parameters
assumptions (4)
- domain assumption Lower bound dim S3(I, lambda) >= (2-2lambda)/(1+lambda) for 1/3 <= lambda <= 3/5, from Beresnevich [2] and Badziahin [1].
- domain assumption Counting estimates (9) and (10): the height bound on a and the bound on the number of vectors q sharing an interval J(m), from [1, equations (20)-(21)].
- domain assumption [1, Lemma 6], [1, Lemma 9], and [1, Proposition 5] on Mobius normal forms and separated roots of cubic polynomials.
- standard math Jarnik-Besicovitch theorem, Minkowski's second theorem, Davenport's class-number estimate, Cauchy's root bound, and Gelfond's lemma.
Cite this review
Pith. "Pith review of Simultaneous Diophantine approximation on the three dimensional Veronese curve." pith.science (2026). https://pith.science/paper/N7V6JLJB
@misc{pith2026250721401,
author = {Pith},
title = {Pith review of: Simultaneous Diophantine approximation on the three dimensional Veronese curve},
year = {2026},
howpublished = {\url{https://pith.science/paper/N7V6JLJB}},
note = {Machine review of arXiv:2507.21401}
}
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.
Reference graph
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