Published paper

Badly approximable systems of affine forms and incompressibility on fractals

Ryan Broderick, Lior Fishman, David Simmons

Published version ↗Published PDF ↗Open DOI ↗arXiv record →

Publication metadata

Journal citation: Journal of Number Theory 133(7), 2186–2205 (2013)
Publication date: 2013-03-05
DOI: 10.1016/j.jnt.2012.12.004 ↗
arXiv counterpart: arXiv:1208.2091v1

Abstract

The paper proves hyperplane-absolute-winning and incompressibility results for badly approximable systems of linear and affine forms.

Role in dependence graphs

Proof-critical source

On Some Properties of Irrational Subspaces

This paper is included only for the following marked statement:

HAW theorem for badly approximable systems · Theorem 1.3Upgrades the relevant set of systems of linear forms to hyperplane absolute winning.

Proof-critical source

Metric theory with a fixed matrix

This paper is included only for the following marked statement:

Theorem 4.1 · §4, theorem and complete proofFor a lacunary sequence of matrices and uniformly discrete targets, the escaping set is hyperplane absolute winning. The focal Proposition 7.7 is its vector-valued special case.

AI-generated audit

Audit summary

Audited against an explicitly disclosed arXiv fallback

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Exact reviewed source

arXiv:1208.2091v1 · explicit fallback for inaccessible version of record

Ryan Broderick, Lior Fishman, David Simmons. Badly approximable systems of affine forms and incompressibility on fractals. arXiv:1208.2091v1.

The Journal of Number Theory version of record was access-controlled and could not be retrieved for this audit; the report therefore evaluates only the exact arXiv v1 manuscript and does not claim to audit the VOR.

Open audited source ↗

Generated August 23, 2026

01Statements4 reported findingsContains unsupported statements

The lacunary escaping-set theorem, the fixed-matrix theorem, its incompressibility corollary, and the negative diffuse-set example are correct after verified repairs to three internal proof defects. The hyperplane-absolute-winning theorem for homogeneous systems, and the fixed-shift corollary that builds on it, are not able to be fully verified from the available exact sources because their proof chain materially imports Schmidt's inaccessible 1969 lemmas; no counterexample to either statement was found.

Theorems 4.1 and 1.1Correct

Lacunary escaping sets and fixed-matrix affine slices are HAW

Pages 2 and 7–8 · Theorems 1.1 and 4.1 and proof · arXiv:1208.2091v1

For each norm window, lacunarity leaves at most $n$ dangerous indices. Each corresponding inverse image of a target ball lies in a hyperplane neighborhood of thickness at most $\beta^r\rho(B_j)$ after replacing the printed affine-plane definition by the verified scalar-projection definition recorded in the proofs section. Repeating the percentage deletion for $r=\lfloor\log_2 n\rfloor+1$ moves eliminates all dangerous neighborhoods and gives a uniform positive avoidance constant. Lemma 2.1 then transfers the strategy to the hyperplane absolute game. For an irrational fixed matrix, the exact cited Cambridge article supplies a lacunary sequence $Y$ with $\widetilde E(Y,\mathbb Z)\subseteq\operatorname{Bad}_A(M,N)$ . In the complementary rational case, that source shows that $\operatorname{Bad}_A(M,N)$ contains the complement of a countable family of parallel affine hyperplanes, which is HAW by the elementary one-hyperplane strategy and countable-intersection stability. Upward closure of HAW proves Theorem 1.1 in both cases.

Exact arXiv v1 manuscript ↗

Corollary 1.2 and Example 3.2Correct

The incompressibility consequence and the diffuse-set obstruction are correct

Pages 2 and 6 · Corollary 1.2, Proposition 3.1, and Example 3.2 · arXiv:1208.2091v1

The cited HAW permanence results give countable-intersection stability, nonsingular $C^1$ invariance, winning on hyperplane-diffuse sets, and full dimension on Ahlfors-regular supports, which yield Corollary 1.2 with its stated quantifiers. In Example 3.2, the three similarities satisfy the open set condition and have noncollinear fixed points. The recursive separation gives an absolute slope bound of $5$ on pairs of points in the limit set, so the set lies on a globally extended $5$ -Lipschitz graph; the shear $\Phi(x,y)=(x,y+f(x))$ is bi-Lipschitz and maps the horizontal axis over that graph. This proves the claimed obstruction.

Cambridge HAW permanence source ↗

Theorem 1.3Not able to verify

The homogeneous-system HAW conclusion has an unresolved source obligation

Pages 2 and 9–14 · Theorem 1.3, Propositions 5.1–5.2, and Sections 5–7 · arXiv:1208.2091v1

The determinant-avoidance construction, after the verified repairs recorded in the proofs section, reduces the theorem to Propositions 5.1 and 5.2. Those propositions are explicitly Schmidt's Lemmas 1 and 2 from the 1969 Journal of Number Theory article and supply the dimension bounds that place each family of dangerous integer solutions in the required finite-dimensional subspace. The exact cited version of that article could not be obtained, so this proof-critical input could not be checked. This is an evidentiary limitation, not evidence that either imported lemma or Theorem 1.3 is false.

Publisher record for Schmidt's 1969 article ↗

Corollary 1.4Not able to verify

The fixed-shift HAW assertion is not established in the reviewed version

Page 2 · Corollary 1.4 and the sentence immediately following it · arXiv:1208.2091v1

The manuscript states that the fixed-shift result follows by combining additional steps from Einsiedler–Tseng with the proof of Theorem 1.3, but it omits that adaptation. The cited paper proves a Schmidt-winning result, whereas upgrading the modified construction to a hyperplane-absolute strategy requires substantive compatibility checks that are not supplied here. The route also inherits the unresolved Schmidt-lemma obligation in Theorem 1.3. No counterexample was found, but the available material does not provide a complete verification.

Einsiedler–Tseng cited source ↗

02Proofs7 reported findingsContains incorrect or incomplete proofs

Three proof arguments are incorrect as written: Theorem 4.1 defines an affine preimage that can be empty, Lemma 7.2 equates a largest minor with the Euclidean norm of all minors, and the induction proving Lemma 6.1 uses a nonexhaustive case split and false norm/diameter constants. All three have complete verified repairs that preserve the advertised statements. The proof of Theorem 1.3 still has an unverified exact-source dependency, and Corollary 1.4 omits a substantive adaptation. Four mechanical notation corrections are harmless.

Theorem 4.1, Equations (4.17)–(4.19)Incorrect as written · verified repair

The printed inverse-image affine hyperplane can be empty

Page 8 · proof of Theorem 4.1, paragraph between Equations (4.18) and (4.19) · arXiv:1208.2091v1

The target point $y_k\in Z_k$ need not lie in $\operatorname{im}(M_k)$ , so $M_k^{-1}(y_k)$ can be empty and the printed set $L=M_k^{-1}(y_k)+W$ is then not an affine hyperplane. Define instead $L=\left\{x\in\mathbb R^N:\left\langle M_kx-y_k,M_kv_k\right\rangle=0\right\}.$ Its normal is $M_k^\top M_kv_k$ , and $\lVert M_k^\top M_kv_k\rVert\geq\langle M_k^\top M_kv_k,v_k\rangle=t_k^2$ . Consequently, if $\lVert M_kx-y_k\rVert<c$ , then $\operatorname{dist}(x,L)\leq\frac{c\,t_k}{\lVert M_k^\top M_kv_k\rVert}\leq\frac{c}{t_k}.$ This proves exactly the containment needed in (4.19) for every target point, including points outside the image, and the remainder of the percentage-game argument is unchanged. Repair classification: Verified repair.

Exact arXiv v1 manuscript ↗

Lemma 7.2Incorrect as written · verified repair

A largest minor is incorrectly identified with the Euclidean norm of all minors

Pages 12–13 · Lemma 7.2 and its proof · arXiv:1208.2091v1

After choosing $\omega'$ to maximize $|D_{\omega'}(A)|$ , the proof uses $|D_{\omega'}(A)|=M_{v-1}(A)$ . With the manuscript's Euclidean norm this equality is generally false. Since $|\Omega_{v-1}|=\binom{N}{v-1}^2$ , the valid bound is $|D_{\omega'}(A)|\geq\frac{M_{v-1}(A)}{\binom{N}{v-1}}.$ Put $d_v=\binom{N}{v-1}$ . The displayed directional-derivative calculation then gives at least $(d_v^{-1}-\sqrt N\,\varepsilon_1)M_{v-1}(A)$ . Choosing $\varepsilon_1\leq(2\sqrt N\,d_v)^{-1}$ and $\varepsilon_2\leq[2d_v(1+\sqrt N\,\sigma)]^{-1}$ proves the lemma with the required uniform positive constants. Every later use needs only their positivity. Repair classification: Verified repair.

Exact arXiv v1 manuscript ↗

Lemma 6.1 inductionIncorrect as written · verified repair

The case split and ball-diameter estimates are invalid as printed

Pages 13–14 · Claim 7.3 and Cases 1–2 in the proof of Lemma 6.1 · arXiv:1208.2091v1

The two cases are stated on the original ball $B$ , but Case 2 later needs its witness $A$ to lie in $B_{v-1}$ in order to bound $\lVert A'-A\rVert$ ; thus the printed cases do not support the subsequent Taylor estimate. Moreover, two points of a ball of radius below $\mu_{v-1}\rho_B$ can be $2\mu_{v-1}\rho_B$ apart, not $\mu_{v-1}\rho_B$ apart as used in Claim 7.3 and the quadratic remainder. A complete repair is to split the cases on $B_{v-1}$ and put $d_v=\binom{N}{v-1}$ and $C_v=1+2d_v(v-1)$ . Applying the mean-value estimate componentwise to the $d_v^2$ minors and then the Euclidean triangle inequality gives $M_{v-1}(B_{v-1})\leq C_vM_{v-1}(A)$ . In Case 2, choose $\mu_{v-1}=\frac{\beta^2\varepsilon_2\nu_{v-1}}{4v^2};$ the first-crossing property gives $\rho(B_{v-1})\geq\beta\mu_{v-1}\rho_B$ , so the linear term is at least $\beta^2\varepsilon_2\mu_{v-1}\rho_BM_{v-1}(A)$ , while the corrected quadratic remainder is at most half of it. Taking $\nu_v$ no larger than both $\varepsilon_1/C_v$ and $\beta^2\varepsilon_2\mu_{v-1}/(2C_v)$ proves (6.33) in both cases; the unnecessary printed $\sigma$ denominator in Case 1 is simply removed because $\rho_B<1$ . Repair classification: Verified repair.

Exact arXiv v1 manuscript ↗

Propositions 5.1–5.2Not able to verify

The imported Schmidt dimension bounds could not be checked in the exact cited source

Page 9 · Propositions 5.1–5.2; used on pages 10–12 in Lemmas 5.3–5.4 and Theorem 1.3 · arXiv:1208.2091v1

These propositions are the only input that bounds the spans of the dangerous integer vectors by $N$ and $M$ , respectively. The later determinant strategy uses those bounds decisively to choose the orthonormal systems on which Lemma 6.1 operates. The manuscript cites Schmidt's Lemmas 1 and 2 in the 1969 Journal of Number Theory paper, but the exact article could not be retrieved and no legitimate exact scan was available. Repair classification: No repair supplied; the source statements must be checked in that cited version or independently reproved. This does not show that they are false.

Publisher record for the inaccessible cited article ↗

Corollary 1.4Incomplete as written

The HAW adaptation for a fixed inhomogeneous shift is omitted

Page 2 · Corollary 1.4 and following sentence · arXiv:1208.2091v1

The proof consists only of the assertion that additional steps from Einsiedler–Tseng can be combined with the proof of Theorem 1.3. The cited winning strategy is not itself a hyperplane-absolute strategy, and the manuscript does not identify the modified dangerous families or verify that each is captured by the single hyperplane deletions required here. This is a substantive construction, not an immediate formal consequence of the two cited arguments. Repair classification: No repair supplied.

Einsiedler–Tseng cited source ↗

Lemma 2.1 and Theorem 4.1 strategyCorrect and complete

The percentage-to-absolute conversion and window deletion are otherwise complete

Pages 4–5 and 7–8 · Lemma 2.1 and proof of Theorem 4.1 · arXiv:1208.2091v1

Repeating one percentage move for $m$ turns leaves at most a proportion $(1-p')^m\leq1-p$ of its neighborhoods, and the finite-cover argument converts a percentage move into one legal absolute deletion. In Theorem 4.1, a norm window contains at most $n$ indices, while $r=\lfloor\log_2 n\rfloor+1$ successive half-deletions leave fewer than one dangerous hyperplane. After the affine-plane repair above and the radius correction below, all neighborhood widths are legal and the uniform constant $c$ is independent of the window.

Exact arXiv v1 manuscript ↗

Four local notation correctionsTypo

A radius factor, two indices, and one sign need mechanical correction

Page 7 · Equation (4.15) and paragraph after (4.14); page 8 · paragraph before (4.19); page 13 · proof of Lemma 7.2 · arXiv:1208.2091v1

First, (4.15) must read $\rho_1<\beta^r\delta/(4t_1)$ , not $\rho_1<\beta^r\delta t_1/4$ ; the reciprocal is forced by the next estimate $\delta\beta^{rj}/(2t_1)\geq2\rho(B_j)$ . Second, the sentence after (4.14) should say that $t_k$ lies in the $j$ th window, not the $k$ th window. Third, in the decomposition $x=w+\eta v_k$ the conclusion is $|\eta|>c/t_k$ , not $\eta>c/t_k$ . Fourth, the sum defining $A'e_{i_0}$ in Lemma 7.2 is indexed by $i$ , so its displayed lower limit must be $\sum_{i=1}^N$ , not $\sum_{j=1}^N$ . Each correction is uniquely determined by the surrounding calculation and changes no conclusion.