A height-moment criterion for recovering escaped arithmetic quantum mass

Definitions and mathematical statement

The canonical text is a transition after Conjecture 39: it reports that E. Lindenstrauss proved "this case" in [120], except for escape of limiting mass to a cusp in the noncompact case, and introduces an intermediate problem suggested by Lindenstrauss. The inserted words "OPEN PROBLEMS 17" are a running header. The precise conjecture, meaning of "this case", and following intermediate problem are absent.

According to the supplied aid, the intended background is arithmetic quantum unique ergodicity for normalized Hecke--Maass eigenfunction measures $|\phi_i|^2\,dV$, with $dV$ normalized to total mass one. It reports convergence to $dV$ for compact congruence arithmetic surfaces, and, in the noncompact setting of the cited result, only local limits $c\,dV$ with $0\leq c\leq1$. The missing assertion is $c=1$. The aid attributes a later nonescape result for Hecke--Maass forms on the modular surface to Soundararajan and says that his argument was stated to apply to congruence subgroups. These scope and literature claims are supplied context, not a newly verified status determination.

The aid's companion criterion can be stated independently. Let $X$ be a locally compact metric space, let $H:X\to[1,\infty)$ be continuous and proper, let $\mu_i,\nu$ be Borel probability measures, and assume \[ \mu_i\longrightarrow c\nu\text{ vaguely},\quad 0\leq c\leq1, \qquad \sup_i\int_X H^p\,d\mu_i\leq M<\infty \quad(p>0). \] Then the measures are tight, $c=1$, $\mu_i$ converge weakly to $\nu$ as probabilities, and $\int H^p\,d\nu\leq M$. To formulate the reported rate, let $0\leq q<p$, $|f|\leq L H^q$, and choose continuous cutoffs $\chi_R$ equal to one on $\{H\leq R\}$, zero on $\{H\geq2R\}$, and between zero and one. If, for $R\geq1$, \[ \left|\int\chi_R f\,d\mu_i-\int\chi_R f\,d\nu\right| \leq D\varepsilon_i R^a,\qquad a\geq0,\quad \varepsilon_i\to0^+, \] then \[ \left|\int f\,d\mu_i-\int f\,d\nu\right| \leq D\varepsilon_i R^a+2LMR^{-(p-q)} =O\!\left(\varepsilon_i^{(p-q)/(a+p-q)}\right) \] after choosing $R$ of order $\varepsilon_i^{-1/(a+p-q)}$. The compact-test estimate and uniform moment are assumptions, not estimates proved here for arithmetic eigenfunctions.

For the cusp normalization mentioned by the aid, a cusp with coordinate $0\leq x<w_j$, $y\geq Y_j$, and hyperbolic area $dx\,dy/y^2$ on a surface of total area $A$ has normalized tail mass $w_j/(AR)$ above height $R\geq Y_j$. Its height-$p$ integral is \[ \frac{w_j}{A}\int_{Y_j}^{\infty}y^{p-2}\,dy =\frac{w_jY_j^{p-1}}{A(1-p)}\quad(0<p<1), \] and diverges for $p\geq1$. Summing these expressions over disjoint cusps gives the corresponding cusp contributions; compact-core contributions depend on the chosen height extension. Thus ordinary cusp height only permits finite target moments with $p<1$. These coordinate conventions clarify the reported companion discussion and do not reconstruct the missing intermediate problem.

Short English statement

This fragment identifies escape of mass as the remaining issue in a cited arithmetic quantum-limit result. A supplied companion criterion says that a uniform positive moment of a proper height prevents escape and can turn compact-test estimates into full convergence rates. Neither the surrounding conjecture nor Lindenstrauss's following intermediate problem is actually stated in the record.

Sources

• Local UnsolvedMath registry, permanent ID 20003328, exact source code catalogue entry; aim-other-notes.json, notes[134], transition after Conjecture 39.
• American Institute of Mathematics, Emerging applications of measure rigidity, https://aimath.org/WWN/measrigid/measrigid.pdf. The source attributes the cited case and the suggested intermediate problem to E. Lindenstrauss and uses bibliography label [120], whose full entry is not supplied. Original remarks and literature fields are empty.
• Supplied research attempt 1: context_only, problem status not_a_problem, rigor label proved, novelty confidence low. It reports the Lindenstrauss/Soundararajan arithmetic QUE qualifications and the height-moment reduction. Machine-generated, not peer reviewed, requiring independent verification before citation. No complete Soundararajan reference or exact eigenfunction-family hypotheses are supplied.