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1. Random walks on direct products of groups

   https://doi.org/10.4171/JEMS/1550  

   Golsefidy, Alireza Salehi; Srinivas, Srivatsa (November 2024, Journal of the European Mathematical Society)

   Suppose\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})is generated by a symmetric setSof cardinalitynwherepis a prime number. Suppose the Cheeger constants of the Cayley graphs of\operatorname{SL}_{2}(\mathbb{F}_{p})with respect to\pi_{L}(S)and\pi_{R}(S)are at leastc_{0}, where\pi_{L}and\pi_{R}are the projections to the left and the right components of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p}), respectively. Then the Cheeger constant of the Cayley graph of\operatorname{SL}_{2}(\mathbb{F}_{p})\times \operatorname{SL}_{2}(\mathbb{F}_{p})with respect toSis at leastcwherecis a positive number which only depends onnandc_{0}. This gives an affirmative answer to a question of Lindenstrauss and Varjú. 

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2. Some sharp inequalities of Mizohata–Takeuchi-type

   https://doi.org/10.4171/RMI/1463  

   Carbery, Anthony; Iliopoulou, Marina; Wang, Hong (June 2024, Revista Matemática Iberoamericana)

   Let\Sigmabe a strictly convex, compact patch of aC^{2}hypersurface in\mathbb{R}^{n}, with non-vanishing Gaussian curvature and surface measured\sigmainduced by the Lebesgue measure in\mathbb{R}^{n}. The Mizohata–Takeuchi conjecture states that \int |\widehat{g d\sigma}|^{2} w \leq C \|Xw\|_{\infty} \int |g|^{2} for allg\in L^{2}(\Sigma)and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), whereXdenotes theX-ray transform. As partial progress towards the conjecture, we show, as a straightforward consequence of recently-established decoupling inequalities, that for every\varepsilon>0, there exists a positive constantC_{\varepsilon}, which depends only on\Sigmaand\varepsilon, such that for allR \geq 1and all weightsw \colon \mathbb{R}^{n}\rightarrow [0,+\infty), we have \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\varepsilon} \sup_{T} \Big(\int_{T} w^{(n+1)/2}\Big)^{2/(n+1)}\int |g|^{2}, whereTranges over the family of tubes in\mathbb{R}^{n}of dimensionsR^{1/2}\times \cdots \times R^{1/2}\times R. From this we deduce the Mizohata–Takeuchi conjecture with anR^{(n-1)/(n+1)}-loss; i.e., that \int_{B_R}|\widehat{g d\sigma}|^{2} w \leq C_{\varepsilon} R^{\frac{n-1}{n+1}+ \varepsilon}\|Xw\|_{\infty} \int |g|^{2} for any ballB_{R}of radiusRand any\varepsilon>0. The power(n-1)/(n+1)here cannot be replaced by anything smaller unless properties of\widehat{g d\sigma}beyond ‘decoupling axioms’ are exploited. We also provide estimates which improve this inequality under various conditions on the weight, and discuss some new cases where the conjecture holds. 

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3. Finiteness properties for relatives of braided Higman–Thompson groups

   https://doi.org/10.4171/GGD/731  

   Skipper, Rachel; Wu, Xiaolei (September 2023, Groups, Geometry, and Dynamics)

   We study the finiteness properties of the braided Higman–Thompson groupbV_{d,r}(H)with labels inH\leq B_d, andbF_{d,r}(H)andbT_{d,r}(H)with labels inH\leq PB_d, whereB_dis the braid group withdstrings andPB_dis its pure braid subgroup. We show that for alld\geq 2andr\geq 1, the groupbV_{d,r}(H)(resp.bT_{d,r}(H)orbF_{d,r}(H)) is of typeF_nif and only ifHis. Our result in particular confirms a recent conjecture of Aroca and Cumplido. 

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4. New upper bounds for the Erdős-Gyárfás problem on generalized Ramsey numbers

   https://doi.org/10.1017/S0963548322000293  

   Cameron, Alex; Heath, Emily (March 2023, Combinatorics, Probability and Computing)

   Abstract A$$(p,q)$$-colouring of a graph$$G$$is an edge-colouring of$$G$$which assigns at least$$q$$colours to each$$p$$-clique. The problem of determining the minimum number of colours,$$f(n,p,q)$$, needed to give a$$(p,q)$$-colouring of the complete graph$$K_n$$is a natural generalization of the well-known problem of identifying the diagonal Ramsey numbers$$r_k(p)$$. The best-known general upper bound on$$f(n,p,q)$$was given by Erdős and Gyárfás in 1997 using a probabilistic argument. Since then, improved bounds in the cases where$$p=q$$have been obtained only for$$p\in \{4,5\}$$, each of which was proved by giving a deterministic construction which combined a$$(p,p-1)$$-colouring using few colours with an algebraic colouring. In this paper, we provide a framework for proving new upper bounds on$$f(n,p,p)$$in the style of these earlier constructions. We characterize all colourings of$$p$$-cliques with$$p-1$$colours which can appear in our modified version of the$$(p,p-1)$$-colouring of Conlon, Fox, Lee, and Sudakov. This allows us to greatly reduce the amount of case-checking required in identifying$$(p,p)$$-colourings, which would otherwise make this problem intractable for large values of$$p$$. In addition, we generalize our algebraic colouring from the$$p=5$$setting and use this to give improved upper bounds on$$f(n,6,6)$$and$$f(n,8,8)$$. 

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5. Thermodynamic formalism for dispersing billiards

   https://doi.org/10.3934/jmd.2022013  

   Baladi, Viviane; Demers, Mark F. (January 2022, Journal of Modern Dynamics)

   For any finite horizon Sinai billiard map \begin{document}$ T $$\end{document} on the two-torus, we find \begin{document}$$ t_*>1 $$\end{document} such that for each \begin{document}$$ t\in (0,t_*) $$\end{document} there exists a unique equilibrium state \begin{document}$$ \mu_t $$\end{document} for \begin{document}$$ - t\log J^uT $$\end{document}, and \begin{document}$$ \mu_t $$\end{document} is \begin{document}$$ T $$\end{document}-adapted. (In particular, the SRB measure is the unique equilibrium state for \begin{document}$$ - \log J^uT $$\end{document}.) We show that \begin{document}$$ \mu_t $$\end{document} is exponentially mixing for Hölder observables, and the pressure function \begin{document}$$ P(t) = \sup_\mu \{h_\mu -\int t\log J^uT d \mu\} $$\end{document} is analytic on \begin{document}$$ (0,t_*) $$\end{document}. In addition, \begin{document}$$ P(t) $$\end{document} is strictly convex if and only if \begin{document}$$ \log J^uT $$\end{document} is not \begin{document}$$ \mu_t $$\end{document}-a.e. cohomologous to a constant, while, if there exist \begin{document}$$ t_a\ne t_b $$\end{document} with \begin{document}$$ \mu_{t_a} = \mu_{t_b} $$\end{document}, then \begin{document}$$ P(t) $$\end{document} is affine on \begin{document}$$ (0,t_*) $$\end{document}. An additional sparse recurrence condition gives \begin{document}$$ \lim_{t\downarrow 0} P(t) = P(0) $$\end{document}$. 

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