Joint work with Fregolli and Björklund.
Let \(\alpha \in \mathbb{R}^{2}\), \(\alpha \simeq \tfrac{p}{q} \in \mathbb{Q}^{2}\).
Fix \(c > 0\) and for \(b \in (0,1)\) and \(T >0\): \[\begin{equation} N_{b, T } = \#\{(p,q) \in \mathbb{Z}^{2} \times Z \mid (**) \text{ is satisfied }\} \end{equation}\] where \((**)\) is the condition given by \[\begin{equation} q(q \alpha_{1} - p_{1})(q \alpha_{2} - p_{2}) \in(0,b), |q \alpha_{1} - p_{1}| \leq c, 1 \leq q < T . \end{equation}\] For almost every \(\alpha : N_{b,T}(\alpha) \sim v_{0}( \log T)^{2} b\)
When \(b \gg (\log T)^{-\frac{6}{5} + \varepsilon}\) as \(T \rightarrow \infty\). Furthermore \[\begin{equation} \frac{N_{b,T}(\alpha) - v_{0} (\log T)^{2} b }{ (\log T) b^{\frac{1}{2}}}, \alpha \in [0,1)^{2} \end{equation}\] converges in distribution to the normal law.
Define \[\begin{equation} \Lambda_{\alpha} = \{ (q \alpha - p , q) \mid (p,q) \in \mathbb{Z}^{3}\}. \end{equation}\] Let \(X =\) “space of lattices in \(\mathbb{R}^{3}\), \(\mathrm{covol}= 1\)”. Let \(Y \subseteq X\) be defined as \[\begin{equation} Y = \{ \Lambda_{\alpha} \mid \alpha \in [0,1)^{2}\}. \end{equation}\] We define \[\begin{equation} a(t) = \begin{bmatrix} e^{t} & & \\ & e^{t_{2}} & \\ & & e^{-t_{1} - t_{2}} \end{bmatrix} \end{equation}\] We want to know \(a(t) Y \rightarrow ?\)
\[\begin{equation} \Omega = \{(x,y) \in \underbrace{\mathbb{R}^{3} }_{\text{ for }a \in(0,b)}\mid x_{1}x_{2} \cdot y \in(a,b), |x_{1}|,|x_{2}| \leq C , 1 \leq y < T\}. \end{equation}\] Then when \(a = 0\), \(N_{b,T}(\alpha) = |\Lambda_{\alpha} \cap \Omega|\). For \(n \in \mathbb{N}_{0}\), \[\Omega_{n} = \{(x,y) \in \mathbb{R}^{3} \mid x_{1} x_{2} y \in (a,b), e^{-1} \cdot c < |x_{1}|, |x_{2}| \leq c\}, e^{-n_{1}-n_{2} \leq y < e^{-n_{1} - n_{2}} T }.\]
\[\begin{equation} \Omega = \bigsqcup_{n \in \mathbb{N}_{0}^{2}} a(n)^{-1} \Omega_{n}. \end{equation}\] \[\begin{equation} N_{b,T}(\alpha) = \sum_{n \in \mathbb{N}_{0}^{2}} |\Lambda_{\alpha} \cap a(n)^{-1} \Omega_{n}| \end{equation}\] \[\begin{equation} \int_{[0,1)^{2}}^{} N_{b,T}(\alpha)^{r}\, d\alpha = \sum_{n_{1},\dots n_{r}}^{} \int_{[0,1)^{2}}^{} |\Lambda_{\alpha} \cap a(n_{1})^{-1} \Omega_{n_{r}}| \, d\alpha \end{equation}\]
For \(f_{1},\dots,f_{r} \in \mathcal{C}_{c}^{\infty}(X), t_{1},\dots,t_{r} \in \mathbb{R}^{2}_{ \geq 0}\) \[\begin{equation} \int_{Y}^{} f_{1} \circ a(t_{1}) \dots f_{r} \circ a(t_{r}) = \int_{X}^{} f_{1} \dots \int_{X}^{} f_{r} + O(f_{1},\dots,f_{n}) ( e^{- \delta \min ( \lfloor t_{i}\rfloor, \| \underbrace{t_{i} - t_{j}}_{i \neq j}\| )}) \end{equation}\]
\[\begin{equation} \int_{Y}^{} f_{1} \circ a(t_{1}) \dots f_{r} \circ a(t_{r}) = (\int_{Y}^{} f \circ a(t_{1}) ) \dots ( \int_{Y} f \circ a(t_{r})) + O(f_{1},\dots,f_{n}) ( e^{- \delta \min_{i} ( \| {t_{i} - t_{j}}\| )}) \end{equation}\]
\[\begin{equation} \int_{Y} f \circ a(t_{1},0) = \int_{Y_{1}} f + O_{f}(e^{-\delta t_{1}}) \end{equation}\]
\[\begin{equation} \int_{[0,1)}^{} |\Lambda_{\alpha} \cap a(n)^{-1} \Omega_{n}| \, d\alpha \ll_{a,b} (1 + \|n\|e^{- \lfloor n\rfloor}) \end{equation}\]
This page was updated on January 18, 2026.
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