Winning property of counterexamples to Uniform Littlewood's Conjecture

The first pigeonhole construction in BPV Lemma 1 supplies the proof omitted from An's Lemma 3.1; that line assignment also underlies Corollary 4.2's common-line concentration.

• Lemma 3.1Every rational point admits a primitive rational line with weighted coefficient bounds.
• Corollary 4.2Dangerous rational points in the relevant partition block lie on one rational line.

Kleinbock–Weiss transfers the Euclidean HAW closure and $C^1$ invariance properties to manifolds through its atlas definition.

• Proposition 3.5HAW subsets of a $C^1$ manifold have the required thickness, countable-intersection, and diffeomorphism properties.

The common-line concentration of dangerous rationals is imported from An; NS14 cites the older version's Lemma 4.2, now Corollary 4.2 in v2.

• Lemma 2.1The potential-game strategy partitions dangerous rationals and assigns a common rational line to each block.
• Theorem 1.1$\mathbf{Bad}(s,t)$ is HAW for every nonnegative $s,t$ with $s+t=1$.

NS14 constructs a hyperplane-potential strategy, which FSU's observation and theorem convert to HAW.

• Lemma 2.4The hyperplane-potential deletion family meets the required budget and forces avoidance.
• Theorem 1.1$\mathbf{Bad}(s,t)$ is HAW for every nonnegative $s,t$ with $s+t=1$.

The endpoint weights reduce to one-dimensional badly approximable numbers, whose HAW property follows from BFKRW.

• Theorem 1.1$\mathbf{Bad}(s,t)$ is HAW for every nonnegative $s,t$ with $s+t=1$.

BFK25 adapts the change of variables and multiplicative Dani correspondence from the historically available arXiv v3 of Fregoli–Kleinbock.

• Lemma 4.1The approximation function is converted to the flow parameter used in the fiberwise-divergence correspondence.
• Proposition 4.2Uniform multiplicative approximation is equivalent to fiberwise divergence of the associated diagonal orbit.

Kleinbock–Margulis Lemma 8.3 supplies the unique monotone change of variables used in the multiplicative flow correspondence.

• Lemma 4.1 in arXiv v3The required monotone change of variables exists uniquely.
• Proposition 4.4 in arXiv v3The multiplicative condition is equivalent to the associated diagonal-orbit excursion condition.

Mahler's criterion converts uniform lower bounds on shortest lattice vectors into boundedness of the diagonal orbit in both directions.

• Theorem 2.5Weighted bad approximation is equivalent to boundedness of the associated diagonal orbit.
• Corollary 3.5The matrix-flow result specializes to the weighted two-coordinate correspondence used by the focal paper.

EKL's high-entropy classification imports the root-conditional commutator mechanism, entropy splitting, and product decomposition developed by Einsiedler–Katok.

• Theorem 1.3Positive entropy for the diagonal action forces the algebraic invariance needed in the Littlewood-exception argument.

The leafwise conditional-measure framework supports EKL's classification, while the appendix maximal estimate supports its positive-entropy measure construction.

• Theorem 1.3Positive entropy for the diagonal action forces the algebraic invariance needed in the Littlewood-exception argument.
• Theorem 10.1Positive transversal dimension produces an invariant probability measure with positive entropy.

Margulis–Tomanov turns unipotent invariance into the homogeneous-measure conclusion. EKL restates the theorems and checks their hypotheses, but the original proof-bearing target was inaccessible.

• Theorem 1.3Positive entropy for the diagonal action forces the algebraic invariance needed in the Littlewood-exception argument.

Luzin regularization supplies the large compact continuity sets used to compare conditional measures.

• Theorem 1.3Positive entropy for the diagonal action forces the algebraic invariance needed in the Littlewood-exception argument.

Entropy affinity and the variational principle build a positive-entropy invariant measure from a positive-dimensional compact transversal.

• Theorem 10.1Positive transversal dimension produces an invariant probability measure with positive entropy.
• Theorem 1.5The classical Littlewood exceptional set has Hausdorff dimension zero.

Standard entropy and generator results supply the upper-semicontinuity step in EKL's variational construction.

• Theorem 10.1Positive transversal dimension produces an invariant probability measure with positive entropy.

Ledrappier–Young Theorem C identifies partial-entropy increments with Lyapunov exponents times transverse dimensions, yielding the entropy formula for ordered root factors.

• Lemma 6.2Entropy splits into the weighted sum of root contributions.
• Corollary 7.2Ledrappier–Young's formula applies to the one-dimensional root factors.
• Proposition 8.3Conditional measures on an unstable subgroup factor over the ordered root foliations.

Part II uses the Section 5 subordinate-partition entropy identity for Corollary 7.2.2 and revisits Proposition 5.1 at each Lyapunov layer in Proposition 11.2, closing the terminal partial-entropy step in Theorem C.

• Theorem CMetric entropy decomposes as Lyapunov exponents times transverse conditional dimensions.

Besicovitch covering supplies the bounded-overlap selection in Lindenstrauss's appendix maximal estimate.

• Appendix Theorem A.1The maximal ergodic estimate used in EKL's measure construction holds with the stated constants.

Theorem 2.13 produces the conjugate rational anisotropic maximal torus and compact diagonal orbit used at the start of EKL's compact-support contradiction.

• Corollary 1.4A positive-entropy invariant measure cannot be compactly supported.

Theorems 1.3 and 3.2 turn the algebraic finite-volume support containing the compact diagonal orbit into an equal-block orbit closure and show every strict enlargement is noncompact.

• Corollary 1.4A positive-entropy invariant measure cannot be compactly supported.

Ratner's homogeneous orbit-closure theorem is the topological rigidity input used through Lindenstrauss–Weiss Theorem 3.3 and Appendix A in the proof of Theorem 1.3.

• Theorem 1.3Classifies diagonal orbit closures containing a compact orbit by equal block groups and proves every strict enlargement is noncompact.

Shah's Corollary 2.13 and Proposition 3.2 supply the stabilizer Zariski-closure and rational algebraic subgroup facts used through Lindenstrauss–Weiss Theorem 3.4 and Appendix A.

• Theorem 1.3Classifies diagonal orbit closures containing a compact orbit by equal block groups and proves every strict enlargement is noncompact.

Raghunathan's lattice, closed-orbit, and Mahler compactness results supply the homogeneous-space structure and final noncompactness step in the equal-block classification.

• Theorem 1.3Classifies diagonal orbit closures containing a compact orbit by equal block groups and proves every strict enlargement is noncompact.
• Theorem 3.2Moore ergodicity makes the diagonal action topologically transitive on the finite-volume algebraic support.

Zimmer supplies the Moore-ergodicity and arithmeticity inputs used to obtain a dense diagonal orbit and rational algebraic structure.

• Theorem 1.3Classifies diagonal orbit closures containing a compact orbit by equal block groups and proves every strict enlargement is noncompact.
• Theorem 3.2Moore ergodicity makes the diagonal action topologically transitive on the finite-volume algebraic support.

The cited linear-algebra theorem supplies the irreducibility and field-action step in the compact-orbit Galois analysis.

• Theorem 1.3Classifies diagonal orbit closures containing a compact orbit by equal block groups and proves every strict enlargement is noncompact.

The cited algebraic-group structure result identifies the connected normalizer and centralizer form used in the Lie-algebra reduction.

• Theorem 1.3Classifies diagonal orbit closures containing a compact orbit by equal block groups and proves every strict enlargement is noncompact.

Dani 1984 Theorem 4.3 supplies property (D) in Shah Proposition 2.5, turning the locally finite invariant measure in the proof chain into a finite one.

• Corollary 2.13A dense unipotent-generated orbit forces containment in the stabilizer's Zariski closure.
• Proposition 3.2The minimal rational algebraic closure has unipotent radical and recovers the connected orbit group.

Moore's general Mautner theorem is imported in Shah Proposition 2.7 to construct the normal subgroup whose fixed vectors force ergodicity.

• Corollary 2.13A dense unipotent-generated orbit forces containment in the stabilizer's Zariski closure.
• Proposition 3.2The minimal rational algebraic closure has unipotent radical and recovers the connected orbit group.

The right-translation continuity proof of Dani's Lemma 8.2 is adapted in Shah Lemma 2.10 and yields ergodicity in the selected chain.

• Corollary 2.13A dense unipotent-generated orbit forces containment in the stabilizer's Zariski closure.
• Proposition 3.2The minimal rational algebraic closure has unipotent radical and recovers the connected orbit group.

Raghunathan supplies the direct properness, arithmetic-lattice, radical, and rational algebraic-group facts used in Shah's Proposition 3.2.

• Proposition 3.2The minimal rational algebraic closure has unipotent radical and recovers the connected orbit group.

Dani84 says Theorem 4.3 follows by repeating Dani79 Theorem 3.3's finite-invariant-exhaustion induction; the exact earlier proof was checked.

• Theorem 4.3Every locally finite measure invariant under a unipotent subgroup is exhausted by finite invariant pieces, Shah's property (D).

Mahler's compactness criterion supplies the compact-return input in the recurrence construction that feeds Dani's rank induction.

• Theorem 3.3Provides the finite invariant exhaustion argument reused in Dani 1984 Theorem 4.3.

The individual ergodic theorem is the standard terminal foundation used in Dani's finite-invariant-exhaustion argument.

• Theorem 3.3Provides the finite invariant exhaustion argument reused in Dani 1984 Theorem 4.3.

Moore explicitly imports Dani's Propositions 4.1 and 5.1 for the two final spectral reductions proving the general Mautner theorem.

• Theorem 1.1An eigenvector for a one-parameter subgroup is fixed by the associated ad-compact normal subgroup.

Moore's earlier semisimple spectral results give the alternative final justification for Theorem 5.2 in the proof of the general Mautner theorem.

• Theorem 1.1An eigenvector for a one-parameter subgroup is fixed by the associated ad-compact normal subgroup.

Oseledets supplies the measurable Lyapunov splitting, exponent limits, and tempered coordinate control used throughout the selected Section 5 entropy construction.

• Proposition 5.1 and Corollary 5.3The Section 5 construction proves that the selected increasing partition subordinate to the full unstable foliation captures the full metric entropy.

Mañé's finite-entropy adapted-partition lemma supplies the small-diameter partition used in the entropy comparison.

• Proposition 5.1 and Corollary 5.3The Section 5 construction proves that the selected increasing partition subordinate to the full unstable foliation captures the full metric entropy.

Ledrappier–Strelcyn supplies the measurable increasing partition and subordinate unstable-disk family reused in the selected Ledrappier–Young construction.

• Proposition 5.1 and Corollary 5.3The Section 5 construction proves that the selected increasing partition subordinate to the full unstable foliation captures the full metric entropy.

The stable-manifold chapter supplies the Pesin local-manifold and Lyapunov-chart foundation used in the selected proof.

• Proposition 5.1 and Corollary 5.3The Section 5 construction proves that the selected increasing partition subordinate to the full unstable foliation captures the full metric entropy.

The bounded-overlap covering selection supplies the Section 4 maximal estimate used in controlling exceptional conditional balls.

• Proposition 5.1 and Corollary 5.3The Section 5 construction proves that the selected increasing partition subordinate to the full unstable foliation captures the full metric entropy.

The cited invariant-manifold chapter supplies the uniform regular-block local-disk theorem from which Ledrappier–Strelcyn constructs its subordinate partition.

• Proposition 3.1Constructs the increasing measurable partition subordinate to global unstable disks reused by Ledrappier–Young.

Oseledets applies pointwise ergodic convergence to logarithms of diagonal cocycle entries and identifies it as the only essential external fact in §4.

• Theorems 1–4Supplies the regular Lyapunov splitting and tempered coordinates used in Ledrappier–Young's selected Section 5 argument.

Walters states and proves the modern discrete $L^1$ pointwise theorem actually used by Oseledets, while the historical article retains its narrower smooth-flow scope.

• Modern discrete $L^1$ formRecords the exact modern probability-preserving-map formulation used by Oseledets, distinct from the historical paper's narrower printed scope.

The preceding PNAS note supplies the disjoint first-return tower integral inequality imported in the pointwise ergodic proof.

• Ergodic theoremProves pointwise time-average convergence within its historical smooth invariant-volume flow scope.

Poincaré recurrence makes the first-return transformation defined and invertible outside a null set in Birkhoff's tower construction.

• Recurrence theorem and tower inequalitySupplies the first-return tower integral argument reused in Birkhoff's following ergodic-theorem paper.

Supplies HAW thickness, countable intersections, C1 invariance, and the diffuse-set restriction results used by the two active appendix examples.

• Lemma 2.1HAW thickness, countable intersections, and C1 nonsingular invariance.
• Corollary 1.3The difference between uniform and ordinary Littlewood counterexamples is HAW and full-dimensional.
• Corollary 1.4The difference locus remains HAW after countably many weighted badly approximable conditions.
• Corollary 1.10The dynamical difference locus with countably many bounded weighted orbits is HAW.
• Appendix A, Example 1The complement of the one-dimensional Sierpiński gasket is HAW on every stated curve but is not HAW in R2.
• Appendix A, Example 2An ambient HAW set need not restrict to HAW on a nonzero-curvature C2 curve.

Converts potential winning to HAW and verifies the Euclidean hyperplane-game hypothesis.

• Lemma 2.4Hyperplane potential winning is equivalent to HAW.
• Theorem 1.2The uniform-Littlewood counterexample set E is HAW in R2 and has full Hausdorff dimension.
• Theorem 1.6Every power-rapid sequence Q gives a HAW fixed-sequence set E_Q.
• Appendix A, Example 1The complement of the one-dimensional Sierpiński gasket is HAW on every stated curve but is not HAW in R2.

Makes manifold HAW intrinsic and supplies the local chart/dynamical transfer used in the torus and curve formulations.

• Lemma 2.2Intrinsic manifold HAW properties, including open-submanifold transfer and chart invariance.
• Theorem 1.9The non-fiberwise-divergent locus is HAW on the horospherical two-torus.
• Corollary 1.10The dynamical difference locus with countably many bounded weighted orbits is HAW.
• Theorem 1.11The non-fiberwise-divergent locus is HAW on the stated regular curves in the torus.
• Lemma 7.5D is HAW on the curve parameter line.

Supplies the weighted primitive rational line attached to a dangerous rational point.

• Lemma 3.3Each rational point has an associated primitive weighted rational line.
• Lemma 3.4Determinant/simplex control forces relevant rational points onto one line.

Weighted badly approximable sets, including endpoint weights, are HAW.

• Corollary 1.4The difference locus remains HAW after countably many weighted badly approximable conditions.
• Corollary 1.10The dynamical difference locus with countably many bounded weighted orbits is HAW.
• Appendix A, Example 2An ambient HAW set need not restrict to HAW on a nonzero-curvature C2 curve.

The classical Littlewood exceptional set has Hausdorff dimension zero.

• Corollary 1.3The difference between uniform and ordinary Littlewood counterexamples is HAW and full-dimensional.
• Corollary 1.4The difference locus remains HAW after countably many weighted badly approximable conditions.
• Corollary 1.10The dynamical difference locus with countably many bounded weighted orbits is HAW.

Supplies the uniform multiplicative Diophantine/non-fiberwise-divergence correspondence.

• Theorem 1.9The non-fiberwise-divergent locus is HAW on the horospherical two-torus.

Supplies the weighted Dani correspondence for bounded diagonal orbits.

• Corollary 1.10The dynamical difference locus with countably many bounded weighted orbits is HAW.

Provides the simultaneous/dual transference characterization of Bad(1/2,1/2).

• Lemma 7.4The dual-form set D is exactly the pullback of Bad(1/2,1/2).

Theorem 2.1 gives the curve restriction of Bad(1/2,1/2); Theorem 1.2 gives the corresponding line result under the stated Diophantine condition. These are the two winning inputs used in the curve and line branches.

• Lemma 7.5D is HAW on the curve parameter line.
• Theorem 1.7Both E and E_Q restrict to absolute-winning sets on every stated regular C2 curve.
• Theorem 1.8E and E_Q are absolute winning on Diophantine lines and are empty on rational lines.

Corollary 1.12 upgrades the 1/2-winning chart result to absolute/HAW winning on R.

• Lemma 7.5D is HAW on the curve parameter line.
• Theorem 1.7Both E and E_Q restrict to absolute-winning sets on every stated regular C2 curve.
• Theorem 1.8E and E_Q are absolute winning on Diophantine lines and are empty on rational lines.

Corollary 6.3 gives the irrational-slope projection rigidity used for the Sierpiński-gasket example.

• Appendix A, Example 1The complement of the one-dimensional Sierpiński gasket is HAW on every stated curve but is not HAW in R2.

Proposition 5.7 supplies a positive-dimensional unbounded-scale Furstenberg microset in the Sierpiński-gasket example.

• Appendix A, Example 1The complement of the one-dimensional Sierpiński gasket is HAW on every stated curve but is not HAW in R2.

Proposition 5.4 turns a microset of the strongly separated self-homothetic gasket into a miniset.

• Appendix A, Example 1The complement of the one-dimensional Sierpiński gasket is HAW on every stated curve but is not HAW in R2.

Frostman's lemma supplies a positive-dimensional measure supported away from the countable bad-tangent set.

• Appendix A, Example 1The complement of the one-dimensional Sierpiński gasket is HAW on every stated curve but is not HAW in R2.

ABV imports the rooted-tree criterion for both the curve construction and the parallel line construction in §6.

• Theorem 2.1For a nondegenerate C2 graph and 0<j≤i<1, the parameter pullback of Bad(i,j) is 1/2-winning.
• Theorem 1.2Under the printed Diophantine condition, the pullback of Bad(i,j) to a line is 1/2-winning.

König's lemma extracts a compatible infinite regular subtree from the finite approximants.

• Proposition 2.1The rooted-tree regular-subtree criterion used by ABV.

Potential winning is converted to absolute winning.

• Theorem 1.51-Cantor-winning implies absolute winning through potential winning.
• Corollary 1.12For subsets of R, 1/2-winning, 1-Cantor-winning, and absolute winning are equivalent.

Positionality and finite-chain compatibility are used in the construction proving 1/2-winning implies 1-Cantor-winning.

• Theorem 1.11Every 1/2-winning subset of R is 1-Cantor-winning.
• Corollary 1.12For subsets of R, 1/2-winning, 1-Cantor-winning, and absolute winning are equivalent.

Proposition 3.13 supplies the symbolic-covering upper bound for Assouad dimension used at the start of KOR's construction.

• Proposition 5.7Every compact planar set has an unbounded-scale Furstenberg microset of Hausdorff dimension at least its Assouad dimension.

KR states Proposition 5.4 by direct reference to Hochman's Claim 9.6.

• Proposition 5.4Strongly separated self-homothetic sets are Furstenberg homogeneous.

The proof of Hochman Theorem 1.6 supplies exponential separation for irrational projected gasket IFSs.

• Corollary 6.3Irrational-slope projections preserve Hausdorff dimension of every Borel subset of the one-dimensional Sierpiński gasket.

Frostman's lemma supplies a measure on the arbitrary Borel subset.

• Corollary 6.3Irrational-slope projections preserve Hausdorff dimension of every Borel subset of the one-dimensional Sierpiński gasket.

The mass distribution principle converts the projected Frostman bound into a Hausdorff-dimension lower bound.

• Corollary 6.3Irrational-slope projections preserve Hausdorff dimension of every Borel subset of the one-dimensional Sierpiński gasket.

The small-exponential-doubling structural theorem is a crucial input to Shmerkin's inverse theorem.

• Theorem 1.11The Lq-dimension formula for pleasant dynamically driven self-similar models.
• Theorem 6.2Lq dimension and Frostman bounds for exponentially separated homogeneous self-similar measures.

The asymmetric Balog–Szemerédi–Gowers theorem is the second external additive-combinatorial input.

• Theorem 1.11The Lq-dimension formula for pleasant dynamically driven self-similar models.
• Theorem 6.2Lq dimension and Frostman bounds for exponentially separated homogeneous self-similar measures.

The upper-C-approximable cocycle theorem supplies uniform limsup control in Shmerkin Lemma 4.7.

• Theorem 1.11The Lq-dimension formula for pleasant dynamically driven self-similar models.
• Theorem 6.2Lq dimension and Frostman bounds for exponentially separated homogeneous self-similar measures.

Basic monotonicity and concavity of the Lq spectrum are used in Proposition 4.6 and the multifractal estimates.

• Theorem 1.11The Lq-dimension formula for pleasant dynamically driven self-similar models.
• Theorem 6.2Lq dimension and Frostman bounds for exponentially separated homogeneous self-similar measures.

The subadditive ergodic theorem gives the almost-sure cocycle slope T(q) in Proposition 4.6.

• Theorem 1.11The Lq-dimension formula for pleasant dynamically driven self-similar models.
• Theorem 6.2Lq dimension and Frostman bounds for exponentially separated homogeneous self-similar measures.

Exact dimensionality identifies self-similar measure dimension with entropy dimension in Hochman's proof.

• Theorem 1.1Dimension drop for a self-similar measure forces superexponential cylinder concentration.
• Corollary 1.2Dimension drop for a self-similar set forces superexponential cylinder concentration.
• Theorem 1.6Irrational projections of the one-dimensional Sierpiński gasket have full dimension; its proof supplies the needed separation argument.

The quantitative Gaussian approximation is used in the entropy inverse theorem.

• Theorem 1.1Dimension drop for a self-similar measure forces superexponential cylinder concentration.
• Corollary 1.2Dimension drop for a self-similar set forces superexponential cylinder concentration.
• Theorem 1.6Irrational projections of the one-dimensional Sierpiński gasket have full dimension; its proof supplies the needed separation argument.

Differentiation of measures is used to identify conditional-density limits in the proof chain.

• Theorem 2.8Conformal/self-similar invariant measures are exact dimensional.

Rokhlin disintegration supplies the canonical conditional measures used throughout Feng–Hu §§3–6.

• Theorem 2.8Conformal/self-similar invariant measures are exact dimensional.

The Radon–Nikodym density theorem constructs the fiber densities in the sufficiency proof for the canonical system.

• Canonical conditional-measure theoremEvery measurable partition of a Lebesgue space has an essentially unique canonical system of conditional measures.

The weak-type differentiation estimate proves integrability in Feng–Hu Proposition 3.5.

• Theorem 2.8Conformal/self-similar invariant measures are exact dimensional.

The triangular-array ergodic lemma closes the conditional Shannon–McMillan step.

• Theorem 2.8Conformal/self-similar invariant measures are exact dimensional.

The conditional-expectation limit theorem is used in the Lyapunov-diameter estimates of Proposition 5.6.

• Theorem 2.8Conformal/self-similar invariant measures are exact dimensional.

Walters supplies algebra approximation and the standard pointwise/subadditive ergodic forms invoked in §§3, 5, and 6.

• Theorem 2.8Conformal/self-similar invariant measures are exact dimensional.

The line proof uses the dual-form formulation and the primitive attached-line construction developed in Badziahin–Pollington–Velani.

• Theorem 1.2Under the printed Diophantine condition, the pullback of Bad(i,j) to a line is 1/2-winning.