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arXiv:1311.1560v1

Values of binary quadratic forms at integer points and Schmidt games

Dmitry Kleinbock, Barak Weiss

math.NT11E1637A4537D40
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Paper metadata

Original submission
November 7, 2013
Claimed MSC
11E16, 37A45, 37D40
arXiv categories
math.NT
Comments
13 pages

Abstract

We prove that for any countable set AA of real numbers, the set of binary indefinite quadratic forms QQ such that the closure of Q(Z2)Q(\mathbb{Z}^2) is disjoint from AA has full Hausdorff dimension.

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Audit summary

Audited against arXiv v1

Not a correctness certificate. A “Correct” result may include yellow typos or minor formal corrections 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.

Current report

Detailed mathematical audit

Generated August 19, 2026
01Statements3 reported findingsCorrect

The thickness theorem for bounded diagonal orbits avoiding countably many transversal submanifolds and its binary-quadratic-form consequence are correct. The abstract and one informal consequence must exclude the zero lattice vector when the forbidden set contains zero.

Theorem 1.4Correct

Bounded orbits can avoid the prescribed transversal set

Page 3 and Sections 4--5 · dynamical main theorem and percentage-game proof · arXiv:1311.1560v1

Transversality lets the percentage-game player delete uniformly controlled parameter intervals whose orbit pieces approach a given submanifold. Scheduling the countably many submanifolds and both time directions preserves a fixed positive proportion of legal moves. The winning-to-thickness and slicing arguments then give full Hausdorff dimension in every open subset of the lattice space.

Theorem 1.3Correct

The quadratic-form value-avoidance statement is valid for nonzero lattice points

Page 2 and Section 2 · quadratic-form theorem and dynamical reduction · arXiv:1311.1560v1

For a≠0a\ne0, lattices containing a vector on Q0=aQ_0=a form a countable union of closed horocycles transversal to both horospherical directions. For a=0a=0, boundedness of the diagonal orbit already keeps Q0Q_0 uniformly away from zero on nonzero lattice vectors. Applying the dynamical theorem to all aa in the countable set yields Q0(Λ∖{0})‾∩A=∅,\overline{Q_0(\Lambda\setminus\{0\})}\cap A=\varnothing, exactly as in the formal statement.

Abstract and informal consequenceMinor formal correction

The phrase `integer points' must exclude the origin

Pages 1--2 · abstract and sentence following Theorem 1.3 · arXiv:1311.1560v1

The formal theorem evaluates Q0Q_0 on x∖{0}x\setminus\{0\}, but the abstract and the prose following Theorem 1.3 say that values `at integer points' miss any countable AA. If 0∈A0\in A, those literal formulations are false because Q(0)=0Q(0)=0 for every form. Replacing `integer points' by `nonzero integer points' is a local domain correction consistent with the formal theorem and proof; it changes no substantive result.

02Proofs2 reported findingsCorrect

The value-to-orbit reduction, transversality checks, percentage-game strategy, and slicing argument are correct and complete. The sole domain mismatch is reported as a minor formal correction and does not lower the overall proof status.

Section 2Correct and complete

Accumulation of values is equivalent to meeting the vector-containing horocycles

Pages 4--6 · Proposition 2.1 and reduction to the dynamical theorem · arXiv:1311.1560v1

If orbit lattices converge to a lattice containing v\mathbf v with Q0(v)=aQ_0(\mathbf v)=a, corresponding lattice vectors have values tending to aa. Conversely, a sequence of nonzero lattice vectors with values tending to nonzero aa can be diagonally rescaled to converge to the fixed vector v\mathbf v; boundedness supplies a convergent lattice subsequence. The primitive-vector decomposition makes the target a countable union of the required horocycles.

Sections 4--5Correct and complete

The percentage-game deletions handle countably many targets

Pages 8--12 · reduction to discrete time and completion of the proof · arXiv:1311.1560v1

Local product coordinates reduce each dangerous encounter to an interval whose tangent direction is separated from the playing direction by transversality. At every scheduled scale only a uniformly bounded fraction of children is deleted. The percentage-game lemma converts this survival proportion to a winning set, and countable scheduling ensures avoidance of every target without reducing the dimension conclusion.

03Novelty0 reported findingsNo non-novelty findings

No non-novelty findings.

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Paper
arXiv:1311.1560v1
Authors listed
Dmitry Kleinbock, Barak Weiss
Audit date
August 19, 2026
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