Invariance certificates and obstructions for representation-ball averages

Definitions and mathematical statement

N. Shah's Question 19 asks for the limiting distribution of $\mu_T$ as $T\to\infty$, but the extracted sentence omits the definition of $\mu_T$. The source says special cases are known [53,73] and motivates distribution of $\Gamma$-orbits on $G/H$ when $\Gamma\cap H$ is a lattice in $H$, or when $\Gamma H$ is dense in $G$. These are context, not a complete universal equidistribution hypothesis.

The supplied partial criterion uses a locally compact second-countable group $G$ with left Haar measure $m$, a lattice $\Gamma$, quotient map $\pi:G\to G/\Gamma$, and measurable sets $R_T\subset G$ with $0<m(R_T)<\infty$. Define \[ \lambda_T=\frac{1_{R_T}\,m}{m(R_T)},\qquad\mu_T=\pi_*\lambda_T. \] For every $h\in G$, the aid's sharp group-level estimate and quotient contraction give \[ \|h_*\mu_T-\mu_T\|_{\mathrm{TV}} \leq\frac{m(hR_T\mathbin{\triangle}R_T)}{2m(R_T)}. \] Here $\|\mu-\nu\|_{\mathrm{TV}}=\sup_E|\mu(E)-\nu(E)|$ for probability measures; the factor $1/2$ depends on this convention.

Define the asymptotic left-symmetry set \[ A(R)=\left\{h\in G:\frac{m(hR_T\mathbin{\triangle}R_T)}{m(R_T)}\longrightarrow0\right\}. \] The reported result is that $A(R)$ is a subgroup and every weak probability cluster measure of $\mu_T$ is invariant under $\overline{A(R)}$. If $A(R)$ contains a subset generating a dense subgroup of $G$, and the family $(\mu_T)$ is tight, then \[ \mu_T\Longrightarrow m_{G/\Gamma}, \] where the quotient Haar probability is normalized to mass one. Tightness prevents escape of probability mass; dense asymptotic symmetries alone are not asserted sufficient. The theorem does not require $A(R)=G$. In fact, the aid notes that full-$G$ asymptotic invariance of these normalized Haar sets is a Følner condition and is impossible for a connected noncompact semisimple Lie group, by nonamenability.

For the representation-ball setup alluded to in the aid, it also reports that for a finite-dimensional continuous linear action the orbit map \[ G\longrightarrow V,\qquad g\longmapsto gp \] is proper exactly when the stabilizer of $p$ is compact and the orbit is closed, in the intended Lie-action setting. Compact stabilizer alone is not sufficient. Its example is the binary cubic $p=x^2y$ under $\operatorname{SL}_2({\mathbb{R}})$: the stabilizer is trivial, but a diagonal family drives the cubic toward zero along an unbounded group sequence. The orbit is nonclosed and norm sublevels in $G$ are unbounded. Unboundedness alone is not here identified with infinite Haar volume. The exact representation-ball definition and the properness, volume-growth, balance, mixing, and nonescape assumptions required by broader positive results are not supplied in the extracted question.

The text beginning "4.4. For irrational $\alpha$" is a different subsection. It recalls equidistribution of $\alpha n^2\pmod1$, expresses an expectation of independent-uniform-like statistics for badly approximable $\alpha$, and defines the ordered-pair statistic \[ R_2([a,b],N,\alpha)=\frac1N\#\left\{1\leq i\ne j\leq N: \alpha(i^2-j^2)\in\tfrac1N[a,b]\pmod1\right\},\qquad[a,b]\subseteq[0,1]. \] This expectation and definition are preserved as excluded context. They are not part of Question 19, and no pair-correlation theorem is inferred from the invariance criterion.

Short English statement

The supplied normalized Haar-set criterion bounds quotient displacement by a symmetric-difference ratio. Dense generated asymptotic symmetries plus tightness force Haar convergence, while a compact stabilizer alone does not ensure proper representation balls. The actual omitted definition of $\mu_T$ and the unrelated pair-correlation subsection remain explicit limitations.

Sources

• Local UnsolvedMath registry, permanent ID $20003288$, exact source code catalogue entry; canonical aim-other-notes.json, notes[94], AIM domain Other, registry category Miscellaneous, source tag question.
• AIM workshop "Emerging applications of measure rigidity," Question 19 by N. Shah, at the end of section 4.3: https://aimath.org/WWN/measrigid/measrigid.pdf. The supplied verification identifies all text from subsection 4.4 onward as extraction spillover. No PDF was fetched.
• Source-local labels [53] and [73] refer to known special cases; full bibliographic entries are not supplied. The original literature field is empty, and no authoritative full references for the broader asymptotic hypotheses or the aid's criterion are given.
• Research attempt 1: result partial_result, problem status at run partially_solved, rigor label proved, novelty confidence low; classification PARTIAL-PROGRESS, registry status partially_solved. Created and updated 14 August 2026. These are supplied assessments requiring review.
• Attribution: Ulam AI / UnsolvedMath Contributors, CC BY 4.0. The machine-generated research aid is not peer reviewed and requires independent verification before citation.