Higher-rank arithmetic QUE and a product-joining obstruction

Definitions and mathematical statement

This record crosses a section boundary. Although its extracted first sentence reads as if Conjecture 39 were proved for some higher-rank symmetric spaces, the supplied aid says the preceding record supplies the words "Results toward". The qualified assertion is therefore that L. Silberman and A. Venkatesh proved results toward Conjecture 39, cited as [178]. The aid identifies [178] as a nondegenerate Cartan-invariant microlocal-lift theorem, not full arithmetic quantum unique ergodicity. It distinguishes a later full theorem requiring compact prime-degree division-algebra spaces, Hecke eigenfunctions, and nondegenerate spectral hypotheses. Neither theorem's complete statement is supplied, so these qualifications must not be dropped.

Section 6.2 then begins a different discussion: M. Berry [19] models eigenfunctions of typical chaotic systems as superpositions of plane waves with random amplitudes, phases and directions, motivating Gaussian and independence predictions for eigenfunctions $\phi_i$ as $i\to\infty$, and a following conjecture for generic negatively curved compact Riemannian manifolds. That following conjecture is absent. No probability sampling, normalization, topology of "generic", or precise sense of independence is specified, so the prose is not a complete Gaussian-limit statement.

The supplied companion joining criterion, separate from either missing conjecture, has the following precise form. Let $X,Y$ be compact metric spaces, $\lambda_i$ probability measures on $X\times Y$, and suppose their two marginals converge weakly to probabilities $\mu,\nu$. For continuous real functions $f,g$ set \[ f_0=f-\int_X f\,d\mu,\qquad g_0=g-\int_Y g\,d\nu. \] Then \[ \lambda_i\longrightarrow\mu\otimes\nu \quad\Longleftrightarrow\quad \int_{X\times Y} f_0(x)g_0(y)\,d\lambda_i(x,y)\longrightarrow0 \quad\text{for every }f\in C(X),\ g\in C(Y). \] Marginal convergence alone is insufficient: normalized Lebesgue measure on the diagonal of $[0,1]^2$ has uniform marginals but correlation $1/12$ for $f(x)=x-1/2$, $g(y)=y-1/2$.

For finite centered families $f_1,\ldots,f_r$ and $g_1,\ldots,g_s$, define the real matrix $C_{jk}=\int f_j(x)g_k(y)\,d\lambda(x,y)$. Its detector strength is \[ \sup_{\|u\|_2=\|v\|_2=1} \left|\int\Big(\sum_j u_jf_j\Big)\Big(\sum_kv_kg_k\Big)\,d\lambda\right| =\|C\|_{\mathrm{op}}. \] If $\lambda=c(\mu\otimes\nu)+(1-c)\rho$ with $0\leq c\leq1$ and $\rho$ a probability, a centered correlation defect $\delta=|\int f_0g_0\,d\lambda|>0$ implies \[ c\leq1-\frac{\delta}{\|f_0\|_\infty\|g_0\|_\infty}. \] When $\mu,\nu$ are Haar probabilities this bounds the product-Haar component. No assertion is made that such defects occur for the actual arithmetic or random-wave families in the fragment.

Short English statement

The entry joins qualified progress toward higher-rank arithmetic QUE to a separate introduction to Berry's random-wave predictions. Neither surrounding conjecture is fully stated. Its supplied companion criterion explains that convergent marginals imply joint independence only when all centered product correlations vanish.

Sources

• Local UnsolvedMath registry, permanent ID 20003331, exact source code catalogue entry; aim-other-notes.json, notes[137], end of the Conjecture 39 discussion and beginning of Section 6.2.
• American Institute of Mathematics, Emerging applications of measure rigidity, https://aimath.org/WWN/measrigid/measrigid.pdf. Numerical references [178] (L. Silberman and A. Venkatesh) and [19] (M. Berry) are retained; full bibliography entries are absent.
• Supplied research attempt 1: context_only, problem status not_a_problem, rigor label proved, novelty confidence low. It supplies the missing "Results toward" qualification, the microlocal-lift/later-theorem distinction, and the joining criterion and detectors. Machine-generated research aid, not peer reviewed; independently verify before citation. No complete citation to the later full theorem is supplied.