This is the second day of the WeWork session in Pune organized by Anish. I am listening to Alex talk about some ideas related to Diophantine approximation using automorphic forms.
Consider \(G_{1} \times G_{2} \supset \Gamma\) . Let \(\gamma = ( \gamma_{1},\gamma_{2})\in \Gamma\). We show that \(\Gamma\) projects densely on \(G_{1}\). For \(r \simeq 0\), \(R \rightarrow \infty\) we have \[\begin{equation} (*) \begin{cases} d(\gamma_{1} x m y_{1}) < r\\ d(\gamma_{2}, e) < R \end{cases} \end{equation}\] Want: there is some solution of \(\gamma \in \Gamma\), for almost everywhere \(x_{1},y_{1} \in G_{1}\) whenever \[\begin{equation} \mathrm{vol}(B_{r}^{(1)}) \cdot \mathrm{vol}(B_{R}^{(2)}) \xrightarrow[]{} \infty \end{equation}\]
Some related work on this is
Consider \(C_{1} \subseteq G\)-compact. Let \(\Omega_{r,R} = \{ y \in C_{1} \mid \nexists \gamma \in \Gamma : (*)\}\). Take: \(\Psi_{1,r} \in \mathcal{C}_{c}(G_{1})\), \(\int_{G_{1}}^{} \psi_{1,r} =1 , \mathrm{supp}(\psi_{1,r}) \subseteq B_{r}^{(1)}\). For fixed \(R_{0} > 0\): \[\begin{equation} \psi_{2,R_{0}}\in \mathcal{C}_{c}(G_{2}): \int_{G_{2}}^{} \psi_{2,R_{0}} =1, \mathrm{supp}(\psi_{2,R_{0}}) \subseteq B_{r_{0}}^{(2)}. \end{equation}\] We then take \[\begin{equation} \psi_{r}(g_{1},g_{2}) = \psi_{1,r}(g_{1}) \psi_{2,R_{0}}(g_{2}) \end{equation}\] For \(x \in G, g \in G\) \[\begin{equation} \psi_{x,r}(y) = \sum_{\gamma \in \Gamma}^{} \psi_{r}(x^{-1} \gamma y) \in \mathcal{C}_{c}(\Gamma \backslash G). \end{equation}\]
For \(L>0: x_{2},y_{2} \in B_{L}^{(2)}\), \(f_{2} \in \mathcal{C}_{c}(G_{2}): \mathrm{supp}(f_{2}) \subseteq B_{R}^{(2)}\). \[\begin{equation} \int_{G}^{} \phi_{x,r}(y g )f_{2}(g) \, dg = \pi(f_{2}) \phi_{x,r}(y) \neq 0 \implies \begin{cases} \exists \gamma \in \Gamma : \\ d(\gamma_{1} x_{1}, y_{1}) < r \\ d(\gamma_{2},e) < 2 L + R_{0} + R \end{cases}. \end{equation}\]
If \(\int_{G_{2}}^{} f_{2} = 1,\) \[\begin{equation} \mathrm{vol}( \Omega_{r,2L + R_{0} + R}) \ll_{C_{1}} \|\pi(f_{2}) \phi_{x,r} - 1\|^{2}_{L^{2}(\Gamma \backslash G)}. \end{equation}\]
\[\begin{equation} \|\pi(f_{2}) \phi - \int_{\Gamma \backslash G}^{} \phi \| _2 \ll \|f_{2}\| _p \|\phi\| _2 \end{equation}\] Then \(f_{2} = \frac{\chi_{B_{R}^{(2)}}}{\mathrm{vol}(B_{R}^{(2)})}\) for \(p < 2\) and \(\mathrm{vol}(B_{R}^{(2)})^{- ( 1-\frac{1}{p})} \cdot \mathrm{vol}(B_{R}^{(1)})^{-1/2}\)
Only works for \(p \simeq 2\). \[\begin{equation} \Omega_{r,R} = \{ (x,y) \in C_{1}^{2} : (*)\} \end{equation}\]
We will rewrite the previous lemma.\(r'>0, y_{2},x_{2} \subseteq B_{L}^{(2)}: f_{1} \in \mathcal{C}_{c}(G_{1}) : \mathrm{supp}(f_{1}) \subseteq B_{r'}^{(1)}\) and \(f_{2} \subseteq \mathcal{C}_{c}(G_{2}): \mathrm{supp}(f_{2}) \subseteq B_{R}^{(2)}\). Then, \(f = f_{1} \otimes f_{2}\) and \[\begin{equation} \pi(f_{1}) \pi(f_{2})= \pi(f) \phi_{x,r}(y) \neq 0 \implies \begin{cases} \exists \gamma \in \Gamma : \\ d(\gamma_{1} x, y_{1}) < r+ r' \\ d(\gamma_{2},e) < 2L + R_{0}+ R \end{cases} \end{equation}\]
Now consider \(G_{1},G_{2}\) be semisimple. Let \(K_{1},K_{2}\) be maximal compact in \(G_{1},G_{2}\) respectively. Let \(\Gamma\) be a cocompact lattice.
Take \(\lambda = C^{2}( \Gamma \backslash G) = \oplus_{\pi \in\tilde{G}} m(\pi) \pi\).
For \(\pi \in \tilde{G}\)-spherical, \(\underbrace{v_{\pi}}_{\text{ unique }} \in \mathcal{H}_{\pi}: \|v_{\pi}\|=1, \pi(K)v_{\pi} = v_{\pi}\). Then \[\begin{equation} \phi_{x,v} = \sum_{\pi}^{} \sum_{j=1}^{m(\pi)} \langle \phi_{x,m}, v_{\pi}^{(j)}\rangle v_{\pi}^{(j)} \end{equation}\]
Then \[\begin{equation} \| \lambda(f) \phi_{x,v}\|^{2} = \sum_{\pi} \sum_{{j}}^{} |\langle \varphi_{x,v} , v_{\pi}^{(j)}\rangle|^{2} \langle \pi_{1}(f) v_{\pi}^{(j), \lambda(f) v_{\pi}^{(j)}}\rangle \end{equation}\] \[\begin{equation} = \sum_{\pi,j}^{} |\langle\phi_{x,v}, v_{\pi^{(j)}}\rangle|^{2} |\langle \pi(f) v_{\pi}^{(j)}, v_{\pi}^{(j)}\rangle|^{2} \end{equation}\] \[\begin{align} = & \int_{G}^{} \psi_{r}(x^{-1} y ) \overline{v_{\pi}^{(j)(\Gamma y)}} \, dy \\ = & \pi(\Psi_{r}) \overline{v_{\pi}^{(j)}(\Gamma x )} \\ = \overline{ \pi( \psi_{r}) v_{\pi}^{(j)}, v_{\pi}^{(j)}} \overline{v_{\pi}^{(j)}(\Gamma x)} \\ = \sum_{\pi,j}^{} | \langle \pi(\psi_{r}) v_{\pi}^{(j)}, v_{\pi}^{(j)}\rangle|^{2} \leq \|\pi(\psi_{r})\| \leq 1 \end{align}\]
So \[\begin{equation} \int_{C_{1}}^{} \|\lambda(f) \phi_{x,r}\|\, dx \ll_{C_{1}} \sum_{\pi \in \widehat{G}}^{} m(\pi) | \langle \pi(\underbrace{f}_{f_{1} \otimes f_{2}} ) v_{\pi}, v_{\pi}\rangle|^{2}. \end{equation}\]
Let us take \(G=\mathrm{SL}_{2}(\mathbb{R}) \times \mathrm{SL}_{2}(\mathbb{R})\)
(FHMM)
\[\begin{equation} \sum_{\sigma [1/2 - \sigma, 1/2}^{} m(\pi_{it} \otimes \pi_{s}) \ll_{\varepsilon} (1+T)^{2 \sigma + \varepsilon} \end{equation}\]
\[\begin{equation} \psi_{2,R'} \in \mathcal{C}_{c}(G_{2}) : \int_{G_{2}}^{} \psi_{2,R'} =1, \end{equation}\] \[\begin{equation} G_{2} \text{ acts on } L^{2}(\Gamma \backslash G)-\text{ spectral gap } \implies \|\lambda( \psi_{2,R'})\| < e^{- \delta R'} \end{equation}\]
\[\begin{equation} \| \underbrace{\lambda(\psi_{2,R'} )\lambda(f) }_{f'=\psi_{2,R'} * f}\phi_{x,r} -1 \|^{2} \end{equation}\]
\[\begin{equation} \| \lambda(\psi_{2,R'}) ( \lambda(f) \phi_{x,r} - 1) \|^{2}_{L^{2}( \Gamma \backslash G)} \ll e^{- 2 \delta R'} \| \lambda(f) \phi_{x,r} - 1\|^{2}_{L^{2}(\Gamma \backslash G)} \end{equation}\] \[\begin{equation} \int_{C_{1}}^{} \| \cdots \|^{2} \, dx \ll_{\varepsilon} e^{-2 \delta R'} e^{ \varepsilon R}, R' = c R \ll \|f\| _p \simeq e^{- \delta_{1} R}. \end{equation}\]
\[\begin{equation} \pi(f) \varphi_{x,r}(y) \neq \emptyset, \mathrm{supp}(f_{2}) \subseteq B_{R}^{(2)} \implies d(\gamma_{2}, e) < R . \end{equation}\]
This page was updated on January 18, 2026.
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