Published paper

On badly approximable numbers and certain games

Wolfgang M. Schmidt

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Publication metadata

Journal citation: Transactions of the American Mathematical Society 123(1), 178–199 (1966)
Publication date: 1966
DOI: 10.2307/1994619 ↗

Abstract

The paper introduces Schmidt's game and proves the permanence and dimension properties used by the later winning-set argument for systems of linear forms.

Role in dependence graphs

Proof-critical source

On Some Properties of Irrational Subspaces

This paper is included only for the following marked statement:

Schmidt-game theorems · winning-set construction and permanence resultsIntroduces the game machinery used in the 1969 winning proof.

Proof-critical source

On Nontrivial Winning and Losing Parameters of Schmidt Games

This paper is included only for the following marked statements:

Lemma 8 · printed p. 182Monotonicity along a fixed product alpha beta: decreasing Alice's parameter preserves winning.
Theorem 7 · §13, printed pp. 198–199Every winning set has a positional winning strategy.
Corollary 1 of Theorem 6 · §§11–12, printed pp. 194–197The Hausdorff-dimension lower bound for an (alpha,beta)-winning set used contrapositively in v3 Proposition 2.3.

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Version of record · Transactions of the American Mathematical Society 123(1), 178–199 (1966)

Wolfgang M. Schmidt. On badly approximable numbers and certain games. Transactions of the American Mathematical Society 123(1), 178–199 (1966).

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Generated August 23, 2026

01Statements4 reported findingsCorrect

The permanence results for winning sets, the winning property and winning dimension of badly approximable numbers, the digit-expansion examples, the Hausdorff-dimension bounds, and the positional-strategy theorem are correct. One mechanical contraction-parameter typo in the proof of the countable-intersection theorem does not alter its statement.

Theorems 1–3Correct

Local-isometry invariance, countable intersections, and badly approximable numbers

Journal pages 184–189 · Theorems 1–3 and their proofs · version of record

The local-isometry argument transfers a winning strategy after the required strict parameter shrinkage $\alpha'<\alpha$ with $\alpha'\beta'=\alpha\beta$ . The scheduled-subsequence construction proves that countable intersections of $\alpha$ -winning sets remain $\alpha$ -winning after the contraction parameter is corrected as recorded under Proofs. The rational-avoidance strategy proves that badly approximable numbers are $(\alpha,\beta)$ -winning whenever $2\alpha<1+\alpha\beta$ , and the obstruction for $\alpha>1/2$ gives winning dimension exactly $1/2$ .

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Theorems 4 and 5Correct

The base-expansion winning-dimension examples

Journal pages 189–194 · Theorems 4–5 and proofs · version of record

The scale-window strategy excludes a prescribed digit at infinitely many positions and therefore establishes the stated winning property for numbers that are not normal to a fixed base. The complementary construction for numbers with infinitely many zero digits supplies both the winning lower bound and an explicit Black strategy above $\alpha_g=((g-1)^2+1)^{-1}$ , proving the exact winning dimension.

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Theorem 6Correct

The branching condition yields the stated Hausdorff-dimension lower bound

Journal pages 194–197 · Theorem 6, Lemmas 20–22, and proof · version of record

The hypotheses create an $m$ -ary tree of legal strategy chains whose level- $k$ balls have radius $(\alpha\beta)^{kt}$ and pairwise disjoint interiors. Lemma 20 bounds by two the number of level balls met by a sufficiently small covering ball. Mapping branches to base- $m$ expansions then gives Hausdorff dimension at least $\log m/(t|\log(\alpha\beta)|)$ . The Euclidean and Hilbert-space corollaries follow from the available branching counts.

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Theorem 7Correct

Every winning set admits a positional winning strategy

Journal pages 198–199 · Theorem 7 and proof · version of record

The well-order construction assigns to each current move a canonical compatible history. If the construction reaches a terminal alternative, the current ball is already contained in the target; otherwise the canonical prefixes extracted from any purported losing play form an infinite chain for the original winning strategy. In both cases the positional strategy forces the intersection into the target set.

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02Proofs5 reported findingsCorrect

The central proofs are correct and complete after one uniquely determined typographical correction in the effective contraction parameter used to interleave countably many strategies.

Theorem 1Correct and complete

Uniform local distortion estimates transfer every legal selected move

Journal pages 184–186 · Lemma 10 and proof of Theorem 1 · version of record

Compactness supplies uniform continuity and positive lower bounds for the local metric multiplier. After entering a sufficiently small ball, the two inclusions in Lemma 10 make the pulled-back and pushed-forward balls legal with parameters $\alpha'=\alpha(1-\varepsilon)^2$ and $\beta'=\beta(1-\varepsilon)^{-2}$ . Radii tend to zero, so the unique outcome in the target space is the image of the outcome governed by the original winning strategy.

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Proof of Theorem 2Typo

The induced Black contraction begins with $\beta$ , not $\alpha$

Journal page 187 · proof of Theorem 2, three occurrences of the induced parameter · version of record

For the strategy assigned to $S_\ell$ , successive selected Black moves are $2^\ell$ rounds apart. Relative to the preceding selected White move, the actual contraction is $\beta(\alpha\beta)^{2^\ell-1}$ . The page instead prints $\alpha(\beta\alpha)^{2^\ell-1}$ , including the special cases $\alpha\beta\alpha$ and $\alpha(\beta\alpha)^3$ . Replace the initial $\alpha$ by $\beta$ in all three occurrences. Each $S_\ell$ is $\alpha$ -winning for every Black parameter, so the corrected parameter supplies the required strategy; the displayed subsequence is then a legal play and its outcome lies in every $S_\ell$ . Repair classification: Verified repair. The theorem and its corollary are unchanged.

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Proof of Theorem 3Correct and complete

The rational-separation induction checks all denominator blocks

Journal pages 188–189 · Lemma 15 and proof of the Proposition · version of record

The choice $\alpha\beta\gamma/2\leq(\alpha\beta)^t<\gamma/2$ , where $\gamma=1+\alpha\beta-2\alpha$ , synchronizes the game scale with denominator blocks. Distinct reduced fractions in one block cannot both be dangerous in the current ball, and Lemma 15 moves the next selected ball past the single possible danger interval. Induction over the blocks yields a uniform lower bound $|x-p/q|>\delta q^{-2}$ for every rational $p/q$ .

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Proofs of Theorems 4–6Correct and complete

The digit-window and branching constructions preserve the required scales

Journal pages 189–197 · Sections 8–12 · version of record

For Theorems 4–5, the selected digit positions are separated far enough that only one forbidden cylinder meets the current interval, while the losing strategy repeats a fixed block containing no zero. For Theorem 6, Lemmas 20–22 provide the separation and tree structure needed for the covering estimate. The endpoint choices, radius multipliers, and dimension exponents agree throughout the constructions.

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Proof of Theorem 7Correct and complete

The canonical-history construction exhausts all positional plays

Journal pages 198–199 · proof of Theorem 7 · version of record

The well-ordering selects compatible least predecessors. The finite-chain compactness step constructs an infinite original-strategy chain whenever canonical prefixes continue indefinitely. Applied to a play of the positional strategy, either a current ball is already contained in the target or the extracted chain has target-contained intersection; both alternatives prove the required winning property.

On badly approximable numbers and certain games

Publication metadata

Abstract

Role in dependence graphs

On Some Properties of Irrational Subspaces

On Nontrivial Winning and Losing Parameters of Schmidt Games

Audit summary

Version of record · Transactions of the American Mathematical Society 123(1), 178–199 (1966)

Local-isometry invariance, countable intersections, and badly approximable numbers

The base-expansion winning-dimension examples

The branching condition yields the stated Hausdorff-dimension lower bound

Every winning set admits a positional winning strategy

Uniform local distortion estimates transfer every legal selected move

The induced Black contraction begins with β\betaβ, not α\alphaα

The rational-separation induction checks all denominator blocks

The digit-window and branching constructions preserve the required scales

The canonical-history construction exhausts all positional plays

The induced Black contraction begins with $\beta$ , not $\alpha$