A quantitative Borel-Cantelli estimate for VWA matrices

Definitions and mathematical statement

This record concatenates a final status fragment concerning Conjecture 24 and regular flats with the setup of section 5.1 on Diophantine approximation. It contains no independent conjecture. The initial fragment mentions compact $\Gamma\backslash G$, measures $\nu_T$ summed over regular flats with systole less than $T$, and Hecke-orbit equidistribution work of Y. Benoist and H. Oh, but supplies neither the full preceding statement nor its measure definition.

For the Diophantine setup, use the max norm and nonzero denominators. A vector $y\in{\mathbb{R}}^n$ is $v$-approximable if $|y\cdot q-p|<\|q\|_\infty^{-v}$ for infinitely many $q\in{\mathbb{Z}}^n\setminus\{0\}$, $p\in{\mathbb{Z}}$. It is very well approximable (VWA) if this holds with $v=n+\varepsilon$ for some $\varepsilon>0$. For $A\in M_{m,n}({\mathbb{R}})$ the source's powered-norm convention is \[ \|Aq-p\|_\infty^{m}<\|q\|_\infty^{-n-\varepsilon} \] for infinitely many $q\in{\mathbb{Z}}^n\setminus\{0\}$, $p\in{\mathbb{Z}}^m$, for some $\varepsilon>0$. The superscript $m$ is a power, not an additional coordinate or dimension parameter hidden in the norm.

The aid proves the following ambient tail estimate for every $m,n\geq1$. For each bounded matrix box $B$ and fixed $\varepsilon>0$, define \[ E_{R,\varepsilon}(B)=\{A\in B:\exists q\ne0,\ \|q\|_\infty\geq R, \ \exists p\in{\mathbb{Z}}^m: \|Aq-p\|_\infty^m<\|q\|_\infty^{-n-\varepsilon}\}. \] There is $C=C(B,m,n,\varepsilon)$ such that for all $R\geq1$, \[ \operatorname{Leb}(E_{R,\varepsilon}(B))\leq CR^{-\varepsilon}. \] For a fixed $q$ of max norm $Q$, put $\delta_Q=Q^{-(n+\varepsilon)/m}$. Each row constraint cuts a slab of width at most a constant times $\delta_Q/Q$ in the coordinate where $|q_j|=Q$. The $m$ rows give $O((\delta_Q/Q)^m)$ volume, there are $O(Q^m)$ relevant numerators, and $O(Q^{n-1})$ denominator vectors in a max-norm shell. The resulting shell sum is $O(\sum_{Q\geq R}Q^{-1-\varepsilon})=O(R^{-\varepsilon})$. Constants are for the fixed bounded box and fixed positive $\varepsilon$.

Letting $R\to\infty$ shows the infinite-solution set for fixed $\varepsilon$ is null. Taking the countable union over $\varepsilon=1/j$ and bounded boxes gives ambient Lebesgue-nullity of all VWA matrices. The $m=1$ statement includes the vector normalization. No estimate uniform as $\varepsilon\to0$ is asserted.

The source then contrasts ambient nullity with intrinsic almost-everywhere behavior on lower-dimensional manifolds. It cites [109], also [17], for the vector Sprindžuk conjecture under the appropriate nondegeneracy hypotheses, described through spans of positive-order coordinate derivatives of charts $f:U\subset{\mathbb{R}}^k\to{\mathbb{R}}^n$. The exact differentiability/order and almost-everywhere qualifications require those references. It says the matrix-manifold analogue needs a suitable notion of nondegeneracy. The ambient estimate above supplies no such definition and proves no extremality statement for the induced measure on an arbitrary lower-dimensional matrix manifold.

Short English statement

This entry is context, not a new conjecture. Its supplied estimate proves that VWA matrices are ambiently Lebesgue-null, with a quantitative $O(R^{-\varepsilon})$ tail under the exact powered max-norm convention. That ambient counting result does not imply an almost-everywhere theorem on a lower-dimensional matrix manifold or define its nondegeneracy.

Sources

• Local UnsolvedMath registry, permanent ID $20003299$, source code catalogue entry; canonical location aim-other-notes.json, zero-based notes[105].
• AIM Emerging applications of measure rigidity, final Conjecture-24 status fragment and section 5 "Diophantine analysis," subsection 5.1, https://aimath.org/WWN/measrigid/measrigid.pdf. The fragment names Y. Benoist and H. Oh for Hecke-orbit averages of maximal compact flats.
• Source labels [109] and [17] for the vector Sprindžuk statement; full bibliography entries are absent and are not guessed.
• Research attempt 1 is marked context-only/not-a-problem, proved, and low novelty confidence, despite the registry conjecture/open labels. Its explicit caveat says the aid is machine generated, not peer reviewed, and requires independent verification before citation.