Published paper

On a problem in simultaneous Diophantine approximation: Schmidt's conjecture

Dzmitry Badziahin, Andrew Pollington, Sanju Velani

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Publication metadata

Journal citation: Annals of Mathematics 174(3), 1837–1883 (2011)
Publication date: 2011
DOI: 10.4007/annals.2011.174.3.9 ↗

Role in dependence graphs

Proof-critical source

Winning property of counterexamples to Uniform Littlewood's Conjecture

This paper is included only for the following marked statement:

Lemma 1 · Annals pp. 1854–1855The first pigeonhole construction supplies the weighted small-coefficient rational line used to fill An's omitted proof.

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Annals of Mathematics version of record

Dzmitry Badziahin, Andrew Pollington, and Sanju Velani. On a problem in simultaneous Diophantine approximation: Schmidt's conjecture. Annals of Mathematics 174 (2011), no. 3, 1837–1883.

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Generated August 27, 2026

01Statements1 reported findingNo statement issues found

The selected rational-line lemma is correct and is stronger than the baby lemma imported by An16.

Lemma 1No issue found

The weighted rational line has the asserted coefficient bounds

VOR pp. 1854–1855 · Lemma 1

For each rational point $P=(p/q,r/q)$ , Lemma 1 constructs a primitive integer line through $P$ with $|A|\le q^i$ and $0<B\le q^j$ under its additional slice hypothesis. The first construction already yields the weaker nonzero- $(A,B)$ form needed by An16 and Wu–Yang without that hypothesis.

02Proofs1 reported findingNo proof issues found

The selected portion is an elementary, complete pigeonhole argument.

Proof of Lemma 1No issue found in the reviewed proof

Equal residues in the weighted box produce the required line

VOR pp. 1854–1855

There are more integer pairs in the prescribed weighted rectangle than residue values of $ap-br$ modulo $q$ . Subtracting two pairs with equal residue gives $Ap-Br+Cq=0$ with the printed bounds and $(A,B)\ne(0,0)$ . The later badly-approximable-coordinate argument only rules out $B=0$ and is irrelevant to the recursive edge.

03Novelty0 reported findingsNo novelty concerns found

No non-novelty findings.

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