arXiv:1502.07648v3

Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces

Lior Fishman, Keith Merrill, David Simmons

math.NT

Paper metadata

Original submission: February 26, 2015
Latest revision: September 15, 2017 · version 3
Claimed MSC
arXiv categories: math.NT
Paper license: arXiv non-exclusive distribution license ↗

Abstract

We prove an analogue of a theorem of A. Pollington and S. Velani ('05), furnishing an upper bound on the Hausdorff dimension of certain subsets of the set of very well intrinsically approximable points on a quadratic hypersurface. The proof incorporates the framework of intrinsic approximation on such hypersurfaces first developed in the authors' joint work with D. Kleinbock (preprint '14) with ideas from work of D. Kleinbock, E. Lindenstrauss, and B. Weiss ('04).

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Audited against arXiv v3

Not a correctness certificate. A green “no issues found” result may include yellow suspected typos or minor formal issues that do not affect substantive soundness. It means this audit found no unresolved substantive error under the stated criteria; it does not replace expert scrutiny or formal verification.

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Detailed mathematical audit

Generated September 26, 2026

01Statements1 reported findingNo statement issues found

The Hausdorff-dimension upper bound for very well intrinsically approximable points on quadratic hypersurfaces is correct. Several defects in the printed neighborhood estimates admit a complete verified repair without changing the theorem's hypotheses or conclusion.

Theorem 1.2No issue found

The intrinsic Pollington–Velani bound holds

Pages 2 and 4–9 · Theorem 1.2 and Section 3 · arXiv:1502.07648v3

Let $\beta<\alpha$ . After the repaired quadratic-neighborhood lemma, every ambient hyperplane section satisfies $\nu(B\cap\mathcal N(\mathcal L\cap Z,\varepsilon\rho))\ll\varepsilon^\beta\nu(B)$ . The intrinsic simplex lemma places rational points of denominator at most $\kappa/\rho_n$ in one affine hyperplane on each ball of radius $\rho_n$ . For an approximant with exponent $c'>c$ , minimality of $n$ gives $q>(\kappa/2)\rho_n^{-1}$ , hence $q^{-c'}\ll\rho_n^{c'}\leq\rho_n^c$ for large $n$ . A maximal $\rho_n^c$ -separated cover then has $s$ -cost bounded by $\sum_n \rho_n^{c(s-\delta)+\beta(c-1)},$ which converges whenever $s>\delta-\beta(1-1/c)$ . Letting $\beta\uparrow\alpha$ proves the displayed bound $\delta-\alpha(1-1/c)$ .

Exact arXiv version 3 ↗

02Proofs6 reported findingsSuspected incomplete or incorrect proofs

The final Hausdorff–Cantelli argument is sound once its inputs are repaired, but the printed proofs of Lemmas 3.4 and 3.5 contain false inclusions and omit necessary covering cases. The repairs below are complete and preserve all stated constants up to the paper's coarse multiplicative notation.

Lemma 3.4Suspected incorrect as written · an apparently sound repair

The tangent-plane estimate is applied to points outside the graph domain

Page 5 · proof of equation (3.1) in Lemma 3.4 · arXiv:1502.07648v3

The maximal separated set $A$ is chosen in the ambient set $\mathcal N(B\cap\Gamma,\varepsilon\rho)$ , but the proof then treats $y,z\in\mathbb R^k$ as points of the graph domain $S\subseteq\mathbb R^{k-1}$ , writes $f(y)$ and $f(z)$ , and invokes a path in $S$ joining them. Moreover, $y$ need not lie in $\operatorname{Supp}(\mu)$ , so the absolute-decay estimate for $B(y,\widetilde\rho)$ is not available. Finally, the assertion that all of $B\cap\Gamma$ lies near the tangent plane at one local center is false; only the corresponding local piece can do so. For a verified repair, put $E=\mathcal N(B\cap\Gamma,\varepsilon\rho)\cap\operatorname{Supp}(\mu)$ and take $A\subseteq E$ maximal $\widetilde\rho$ -separated, where $\widetilde\rho=\sqrt\varepsilon\rho/2$ . For each $y\in A$ and each measured point $w\in B(y,\widetilde\rho)\cap E$ , choose graph points $(u,f(u)),(v,f(v))\in B\cap\Gamma$ within $\varepsilon\rho$ of $y,w$ . Their domain coordinates are $O(\widetilde\rho)$ apart. The $K_1$ -quasiconvex path from $u$ to $v$ and the bound $\|f''\|\leq K_2/\rho$ put the local graph piece within $O(\sqrt\varepsilon\,\widetilde\rho)$ of the tangent hyperplane at $(u,f(u))$ ; adding the two endpoint errors has the same order. Absolute $\alpha$ -decay at the support center $y$ , bounded overlap, and the printed iteration then give (3.1) and (3.2), with only the implicit constant changed.

Lemma 3.5Suspected incorrect as written · an apparently sound repair

Signed permutations of one dominant-coordinate patch do not cover a general diagonal quadric

Page 6 · paragraph defining $\widetilde Z$ in the proof of Lemma 3.5 · arXiv:1502.07648v3

After diagonalizing $P(x)=\sum_i c_i x_i^2-b$ , the proof claims $Z=G(\widetilde Z)$ for the signed permutation group $G$ . This is false when the coefficients $c_i$ are unequal, because permuting coordinates need not preserve $Z$ . Replace this assertion by the finite cover indexed by an orthant and a dominant coordinate $j$ : $Z_{\sigma,j}=\{x\in Z:\operatorname{sgn}(x)=\sigma,\ |c_i x_i|\leq|c_jx_j|\text{ for all }i\}.$ Every point belongs to such a patch by choosing $j$ maximizing $|c_jx_j|$ . On each patch solve for the coordinate $x_j$ with its fixed sign. The derivative calculation printed for $j=k$ is unchanged after relabeling and gives $|f_i'|\leq1$ and $|f_{i\ell}''|\ll1/\min_m|x_m|$ . There are only $2^k k$ patches, so the uniform estimate survives the finite union.

Lemma 3.5Suspected incomplete as written · an apparently sound repair

The dyadic shells neither cover the far-from-axis part nor justify the displayed covering inequality

Page 7 · dyadic-shell argument in the proof of Lemma 3.5 · arXiv:1502.07648v3

The shells $R_n=\{\rho_{n+1}\leq\min_i x_i\leq\rho_n\}$ for $n\geq0$ omit points with $\min_i x_i>\rho$ . Also, a maximal separated set in $B\cap R_n\cap\operatorname{Supp}(\mu)$ need not cover the measured set $B\cap\mathcal N(\Gamma(S_n,f),\varepsilon\rho)$ , whose support points can lie outside $R_n$ . A complete repair is as follows. First handle the portion with $\min_i x_i>\rho$ as one bounded local rectangle around $B$ : there $\|f''\|\ll1/\rho$ , so Lemma 3.4 directly gives $O(\varepsilon^\beta\mu(B))$ . For shells satisfying $\rho_n\geq\varepsilon\rho$ , choose a maximal $\rho_n$ -separated set in $B\cap\mathcal N(\Gamma(S_n,f),\varepsilon\rho)\cap\operatorname{Supp}(\mu)$ . These centers are $O(\rho_n)$ from the union $V$ of coordinate hyperplanes; Lemma 3.4, bounded overlap, and absolute decay therefore give $\mu(E_n)\ll(\varepsilon\rho/\rho_n)^\beta(\rho_n/\rho)^\alpha\mu(B)=\varepsilon^\beta2^{-n(\alpha-\beta)}\mu(B).$ The sum converges because $\beta<\alpha$ . The remaining tail $\rho_n<\varepsilon\rho$ lies, after the outer neighborhood is added, in $\mathcal N(V,O(\varepsilon\rho))$ and has measure $O(\varepsilon^\alpha\mu(B))$ , hence $O(\varepsilon^\beta\mu(B))$ . Coordinate-hyperplane points themselves are included in the same last estimate. This proves Lemma 3.5 with the stated uniformity.

Corollary 3.7Suspected typo

The ball radius and neighborhood scale use different variable names

Page 8 · statement of Corollary 3.7 · arXiv:1502.07648v3

The ball is written $B=B(x,r)$ but the neighborhood thickness is $\varepsilon\rho$ . Replace $r$ by $\rho$ in the ball, or replace $\varepsilon\rho$ by $\varepsilon r$ . The proof and the application on page 9 use one common radius, so the correction is uniquely determined and harmless.

Proof of Theorem 1.2Suspected typo

The minimal-denominator index is reversed

Page 8 · final paragraph before the covering conclusion · arXiv:1502.07648v3

If $n$ is minimal with $q\leq\kappa\rho_n^{-1}$ and $\rho_n=2^{-n}$ , minimality gives $q>\kappa\rho_{n-1}^{-1}=\frac{\kappa}{2}\rho_n^{-1},$ not $q\geq\kappa\rho_{n+1}^{-1}$ . The printed version is incompatible with the defining upper bound. The next displayed estimate $q^{-c'}\leq(2/\kappa)^{c'}\rho_n^{c'}$ is exactly the consequence of the corrected $n-1$ inequality, so the proof already proceeds with the intended index.

Section 3.4Correct and complete after the apparently sound repairs above

The Hausdorff–Cantelli cost calculation closes after the repaired neighborhood lemma

Pages 8–9 · proof of Theorem 1.2 · arXiv:1502.07648v3

Maximal separated sets cover the target limsup and have bounded multiplicity under the Ahlfors-regular measure. Corollary 3.8 supplies one affine hyperplane for the relevant rational points on each scale. Applying the repaired Corollary 3.7 with relative thickness $\rho_n^{c-1}$ yields the exponent $c(s-\delta)+\beta(c-1)$ , and its positivity is equivalent to $s>\delta-\beta(1-1/c)$ . No endpoint or quantifier is lost when $c=1$ , and taking arbitrary $\beta<\alpha$ gives the claimed limit. The exact arXiv version was subsequently published in Selecta Mathematica 24 (2018), 3875–3888, DOI 10.1007/s00029-018-0446-7.

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Paper: arXiv:1502.07648v3
Authors listed: Lior Fishman, Keith Merrill, David Simmons
Audit date: September 26, 2026

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Hausdorff dimensions of very well intrinsically approximable subsets of quadratic hypersurfaces

Paper metadata

Abstract

Audit summary

Detailed mathematical audit

The intrinsic Pollington–Velani bound holds

The tangent-plane estimate is applied to points outside the graph domain

Signed permutations of one dominant-coordinate patch do not cover a general diagonal quadric

The dyadic shells neither cover the far-from-axis part nor justify the displayed covering inequality

The ball radius and neighborhood scale use different variable names

The minimal-denominator index is reversed

The Hausdorff–Cantelli cost calculation closes after the repaired neighborhood lemma

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