Twisted Diophantine Approximation I: Asymptotic Theory

Henk explicitly imports the elementary lattice normalization, basis-extension, finite-index, and successive-minima span facts before proving Theorem 1.5; the same normalization is used in the Section 3 proof of Theorem 1.3. The substantive counting and volume arguments are Henk's own.

• Theorem 1.5Upper bound for lattice points in terms of all successive minima.
• Theorem 1.3Minkowski's second theorem, used to compare covolume with the product of successive minima in a subspace.

Henk Lemma 2.1 applied to $CK$ and the sublattice $\lceil2C\rceil\Lambda$ gives fixed-dilation control. Henk Theorem 1.5 gives the successive-minima product upper bound. Theorem 1.3 and its Section 3 proof supply Minkowski's second theorem for the covolume-product step in the focal exterior-algebra argument.

• Lemma 5.1Fixed-dilation lattice count and two-sided successive-minima cube count.
• Lemma 9.1Exterior-product formula for active and arbitrary primitive sublattices.

Dani Theorem 2.14 converts nonsingularity into nondivergence of the forward lattice trajectory, the starting point of Corollary 2.4. Dani Theorem 2.20 converts bad approximability into boundedness of the same trajectory for Remark 1.5. The block conventions agree after the standard permutation/transposition identification.

• Corollary 2.4Divergence of the critical series and nullity of bad targets for nonsingular matrices.
• Remark 1.5Recovers the dyadically regular badly approximable case through the bounded-orbit correspondence.

Mahler's compactness criterion turns each compact return into a uniform positive shortest-vector bound. Bounded distortion over the fixed interval between consecutive sampled flow times then gives a positive lower bound at infinitely many indices.

• Corollary 2.4Divergence of the critical series and nullity of bad targets for nonsingular matrices.

Apply the general theorem with the torus in the supremum metric, $g(r)=r^m$, and $f(r)=r^s$. The transformed ball radius is $r^{s/m}$, exactly the focal function $\varphi=\psi^{s/m}$; conullity of the transformed limsup follows from Theorem 1.1.

• Corollary 3.1Separate sufficient zero and infinity criteria for Hausdorff measure.

The best-approximation separation bounds the number of orbit points in the short interval. More importantly, the proof of Fuchs-Kim Lemma 2.1 shows that convergence of the block-minimum series implies $\sum_s q_s\psi(q_s)<\infty$, exactly the residual summability needed in the reverse implication.

• Proposition 4.1Equivalence of the focal dyadic series and the Fuchs-Kim block series in dimension one.

Equation (6.9) counts the multiples $tq_s$ for which both the denominator and approximation constraints hold. Adding the zero vector and both signs gives the focal lower bound for $N_\alpha(\psi(2^j),2^j)$.

• Proposition 4.1Equivalence of the focal dyadic series and the Fuchs-Kim block series in dimension one.

The generalized Borel-Cantelli lemma gives the displayed lower bound for a limsup measure in terms of finite sums of event measures and pairwise intersections; this turns Proposition 7.3 into positive measure.

• Lemma 7.2Second-moment lower bound for the limsup measure.

The selected triple-ball covers have diameters tending to zero, and convergence of their $s$-power costs is exactly the hypothesis of the Hausdorff-Cantelli lemma.

• Corollary 3.1Separate sufficient zero and infinity criteria for Hausdorff measure.