A quantitative fixed-horocycle bridge to the critical simultaneous limit

Definitions and mathematical statement

The supplied verification identifies this permanent record as explanatory text between Conjecture 20 (Rudnick--Sarnak) and Conjecture 21, not an independent conjecture. Neither neighboring conjecture is substituted. Its setting is a finite-area hyperbolic surface $X=\Gamma\backslash\mathbb H^2$ with cusp width one, meaning $\Gamma\cap U=\{n_m:m\in{\mathbb{Z}}\}$ for $n_t=\begin{pmatrix}1&t\\0&1\end{pmatrix}$. In $Y=\Gamma\backslash PSL(2,{\mathbb{R}})\cong T^1X$ put $a_y=\operatorname{diag}(\sqrt y,1/\sqrt y)$ and $u_y(t)=\Gamma n_ta_y$. Use the surface convention, or the corresponding orbifold version if torsion is allowed. Then $u_y$ has parameter period one and physical horocycle length $y^{-1}$.

The captured context reports that the normalized continuous horocycle measure tends to Liouville probability $m_Y$ as $y\to0^+$, while for fixed $y$ and irrational $\alpha$, $u_y(n\alpha)$ samples that circle by irrational rotation. The samples use $n\alpha$, not $\alpha^n$. These two separate limits do not specify a simultaneous relation between sample count and height.

The aid's quantitative companion fixes a smooth compactly supported observable $f$ and writes $F_y(t)=f(u_y(t))$. Suppose \[ \|k\alpha\|_{{\mathbb{R}}/{\mathbb{Z}}}\geq c|k|^{-\tau}\quad(k\in{\mathbb{Z}}\setminus\{0\}),\qquad c>0, \] and let the integer $s>\tau+1$. The supplied small-divisor estimate is \[ \left|\frac1M\sum_{n=1}^MF_y(n\alpha)-\int_0^1F_y(t)\,{\,\mathrm{d}} t\right| \leq\frac{\zeta(s-\tau)}{c(2\pi)^sM}\,\|F_y^{(s)}\|_{L^1([0,1])}. \] Here the Riemann zeta function comes from summing Fourier modes; no Diophantine bound is being assumed for every irrational $\alpha$. A complementary estimate for bounded variation is \[ \left|\frac1M\sum_{n=1}^MF_y(n\alpha)-\int_0^1F_y\right| \leq C_\tau\operatorname{Var}(F_y)\,c^{-1/(\tau+1)}M^{-1/(\tau+1)}. \] The constant $C_\tau$ is independent of $M,y,F_y$ under these hypotheses.

Let $Hf(g)=\frac{{\,\mathrm{d}}}{{\,\mathrm{d}} r}|_{r=0}f(gn_r)$ be the physical right-horocycle derivative. The exact scaling is \[ F_y^{(s)}(t)=y^{-s}(H^sf)(u_y(t)),\qquad \|F_y^{(s)}\|_1=y^{-s}\int_0^1|(H^sf)(u_y(t))|\,{\,\mathrm{d}} t. \] Defining $E_{\mathrm{hor}}(y,f)=|\int_0^1F_y-\int_Yf\,{\,\mathrm{d}} m_Y|$, the total discrete-to-Liouville error is at most the Fourier bound plus $E_{\mathrm{hor}}(y,f)$, or the bounded-variation bound plus that term. The context supplies $E_{\mathrm{hor}}(y,f)\to0$ but no rate.

At the critical proposed scale $y\asymp M^{-2}$, the elementary bound $\|F_y^{(s)}\|_1\leq y^{-s}\|H^sf\|_\infty$ makes the first error estimate grow like $M^{2s-1}$ rather than tend to zero. This shows insufficiency of that estimate and of separate qualitative limits; it does not prove failure of the actual simultaneous equidistribution. No missing Conjecture 21 or critical-scale theorem is asserted by this context-only record.

Short English statement

This is the paragraph between two conjectures. Its supplied companion bounds separate fixed-horocycle sampling error from continuous horocycle equidistribution error, and displays the height-dependent derivatives explicitly. Those derivatives make the elementary bound ineffective at height comparable to the inverse square of sample count. This is a quantitative gap, not a disproof of simultaneous equidistribution or a replacement conjecture.

Sources

• Local registry: permanent catalogue entry, exact source code catalogue entry; aim-other-notes.json, notes[96]; inherited canonical tag "conjecture".
• AIM Open Problems from the Workshop "Emerging Applications of Measure Rigidity", Section 4.4, https://aimath.org/WWN/measrigid/measrigid.pdf. The supplied audit places this paragraph after Rudnick--Sarnak Conjecture 20 and before Conjecture 21. No independent numbered conjecture or proposer is present.
• Supplied research attempt 1, registry updated 14 August 2026: context_only, not_a_problem, low novelty confidence, machine-generated and not peer reviewed. It supplies the Fourier constant, BV estimate, physical derivative scaling, and two-scale ledger, without a complete authoritative reference for the continuous-horocycle theorem.
• Statement verification reports the exact contextual boundary in the official PDF; registry status nevertheless remains open/OPEN-TRIAGE. Original remarks and literature field are empty.