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Proof-critical dependence graph

On Nontrivial Winning and Losing Parameters of Schmidt Games

A statement-restricted graph for arXiv:2401.00614v3. It traces the uniform-distribution, Schmidt-game, Hausdorff-dimension, and Borel-determinacy results actually imported by the v3 proofs. Theorem 2.6 is marked self-contained. Contextual citations, an earlier bound that is not used, and the different determinacy citation added only in the later journal version are excluded.

Graph scope6 nodes6 proof-critical linksChecked August 23, 2026

Oriented proof graph

Dependence map

Arrows point from a prerequisite toward the paper whose marked statement uses it.

  • Solid arrow: headline proof lineage
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Every visible arrow is documented in the evidence ledger below. A non-book leaf means that no earlier proof-critical source was identified for the marked statement—not that the paper has no other citations.

Evidence ledger

Proof-critical links

Über ein in der Theorie der säkularen Störungen vorkommendes Problem→On Nontrivial Winning and Losing Parameters of Schmidt GamesHeadline lineageVerified

V3 Lemma 4.1 imports uniform distribution of n theta modulo one. It is used first in Lemma 3.5 to choose game scales, then in Proposition 4.2 to average the number of forced binary digits, and again in the proof of Theorem 2.5 to obtain infinitely many admissible d-adic scales.

Citation location: v3 Lemma 4.1, p. 12; Lemma 3.5, pp. 9–10; Proposition 4.2, pp. 13–14; proof of Theorem 2.5, p. 17; bibliography [B09]
  • Theorem 2.2Winning and losing parameter regions for the upper- and lower-binary-digit-frequency sets, including the two additional c=1/2 endpoint regions.
  • Theorem 2.5A nontrivial losing zone for the eventual d-adic badly approximable set.
On badly approximable numbers and certain games→On Nontrivial Winning and Losing Parameters of Schmidt GamesHeadline lineageVerified

V3 Propositions 3.1–3.2 import Schmidt's fixed-product monotonicity and existence of positional strategies. They are essential in Lemma 3.8's symmetry contradiction, which feeds Proposition 4.3 and the endpoint clauses of Theorem 2.2.

Citation location: v3 Propositions 3.1–3.2, pp. 7–8; Lemma 3.8, pp. 10–11; Proposition 4.3, pp. 15–16; Schmidt Lemma 8 and Theorem 7
  • Theorem 2.2Winning and losing parameter regions for the upper- and lower-binary-digit-frequency sets, including the two additional c=1/2 endpoint regions.
On badly approximable numbers and certain games→On Nontrivial Winning and Losing Parameters of Schmidt GamesAncillary claimVerified

Schmidt's lower bound on the Hausdorff dimension of a winning set is applied contrapositively: once Besicovitch supplies the exact dimension, parameter pairs forcing a larger winning-set dimension must be non-winning; Borel determinacy converts the conclusion to losing.

Citation location: v3 discussion immediately before Proposition 2.3, p. 5; Schmidt Corollary 1 of Theorem 6, printed pp. 194–197
  • Proposition 2.3The ancillary Hausdorff-dimension losing-zone bound for the lower digit-frequency set.
On the sum of digits of real numbers represented in the dyadic system. (On sets of fractional dimensions II.)→On Nontrivial Winning and Losing Parameters of Schmidt GamesAncillary claimSelf-contained leaf

The exact binary-entropy dimension of the lower digit-frequency set is the numerical input combined with Schmidt's winning-set dimension bound in Proposition 2.3.

Citation location: v3 p. 4 and paragraph before Proposition 2.3, p. 5; Besicovitch printed pp. 321–330
  • Proposition 2.3The ancillary Hausdorff-dimension losing-zone bound for the lower digit-frequency set.
Determinacy of Schmidt's Game and Other Intersection Games→On Nontrivial Winning and Losing Parameters of Schmidt GamesHeadline lineagePartially verified

The v3 endpoint proof first establishes non-winning or non-losing statements. Because the relevant digit-frequency sets are Borel, determinacy upgrades them to losing or winning. The same upgrade is implicit when Proposition 2.3 states losing rather than merely non-winning.

Citation location: v3 determinacy paragraph p. 3 citing [CFJ23]; proof of Proposition 4.3, pp. 15–16; Proposition 2.3, p. 5Verification note: Exact JSL full text unavailable; exact v3 citation, publisher metadata, and the freely available author manuscript were checked.
  • Theorem 2.2Winning and losing parameter regions for the upper- and lower-binary-digit-frequency sets, including the two additional c=1/2 endpoint regions.
  • Proposition 2.3The ancillary Hausdorff-dimension losing-zone bound for the lower digit-frequency set.
A purely inductive proof of Borel determinacy→Determinacy of Schmidt's Game and Other Intersection GamesHeadline lineageTerminal source

CFJ does not reprove ordinary Borel determinacy; its §2 theorem explicitly imports Martin's theorem that all Borel games are determined in ZFC. Only this branch of CFJ is relevant to the v3 proof.

Citation location: CFJ §2, theorem and reference [6] in the published bibliography; Martin, A purely inductive proof of Borel determinacy, official pp. 303–308Verification note: Martin is a published proceedings chapter, so no deeper graph edge is drawn.
  • Borel-game determinacy theoremAll Borel games on a nonempty move set with a tree of legal plays are determined in ZFC; the Borel Schmidt games used in the endpoint argument are an instance.