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1. CUTTING A PART FROM MANY MEASURES

   https://doi.org/10.1017/fms.2019.33  

   BLAGOJEVIĆ, PAVLE V.; PALIĆ, NEVENA; SOBERÓN, PABLO; ZIEGLER, GÜNTER M. (January 2019, Forum of Mathematics, Sigma)

   Holmsen, Kynčl and Valculescu recently conjectured that if a finite set $$X$$ with $$\ell n$$ points in $$\mathbb{R}^{d}$$ that is colored by $$m$$ different colors can be partitioned into $$n$$ subsets of $$\ell$$ points each, such that each subset contains points of at least $$d$$ different colors, then there exists such a partition of $$X$$ with the additional property that the convex hulls of the $$n$$ subsets are pairwise disjoint. We prove a continuous analogue of this conjecture, generalized so that each subset contains points of at least $$c$$ different colors, where we also allow $$c$$ to be greater than $$d$$ . Furthermore, we give lower bounds on the fraction of the points each of the subsets contains from $$c$$ different colors. For example, when $$n\geqslant 2$$ , $$d\geqslant 2$$ , $$c\geqslant d$$ with $$m\geqslant n(c-d)+d$$ are integers, and $$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$$ are $$m$$ positive finite absolutely continuous measures on $$\mathbb{R}^{d}$$ , we prove that there exists a partition of $$\mathbb{R}^{d}$$ into $$n$$ convex pieces which equiparts the measures $$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{d-1}$$ , and in addition every piece of the partition has positive measure with respect to at least $$c$$ of the measures $$\unicode[STIX]{x1D707}_{1},\ldots ,\unicode[STIX]{x1D707}_{m}$$ . 

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2. COUNTING CONJUGACY CLASSES IN

   https://doi.org/10.1017/S0004972718000047  

   HULL, MICHAEL; KAPOVICH, ILYA (June 2018, Bulletin of the Australian Mathematical Society)

   We show that if a finitely generated group $$G$$ has a nonelementary WPD action on a hyperbolic metric space $$X$$ , then the number of $$G$$ -conjugacy classes of $$X$$ -loxodromic elements of $$G$$ coming from a ball of radius $$R$$ in the Cayley graph of $$G$$ grows exponentially in $$R$$ . As an application we prove that for $$N\geq 3$$ the number of distinct $$\text{Out}(F_{N})$$ -conjugacy classes of fully irreducible elements $$\unicode[STIX]{x1D719}$$ from an $$R$$ -ball in the Cayley graph of $$\text{Out}(F_{N})$$ with $$\log \unicode[STIX]{x1D706}(\unicode[STIX]{x1D719})$$ of the order of $$R$$ grows exponentially in $$R$$ . 

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3. Almost Mathieu operators with completely resonant phases

   https://doi.org/10.1017/etds.2018.133  

   LIU, WENCAI (July 2020, Ergodic Theory and Dynamical Systems)

   Let $$\unicode[STIX]{x1D6FC}\in \mathbb{R}\backslash \mathbb{Q}$$ and $$\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})=\limsup _{n\rightarrow \infty }(\ln q_{n+1})/q_{n}<\infty$$ , where $$p_{n}/q_{n}$$ is the continued fraction approximation to $$\unicode[STIX]{x1D6FC}$$ . Let $$(H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}u)(n)=u(n+1)+u(n-1)+2\unicode[STIX]{x1D706}\cos 2\unicode[STIX]{x1D70B}(\unicode[STIX]{x1D703}+n\unicode[STIX]{x1D6FC})u(n)$$ be the almost Mathieu operator on $$\ell ^{2}(\mathbb{Z})$$ , where $$\unicode[STIX]{x1D706},\unicode[STIX]{x1D703}\in \mathbb{R}$$ . Avila and Jitomirskaya [The ten Martini problem. Ann. of Math. (2), 170 (1) (2009), 303–342] conjectured that, for $$2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$$ , $$H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$$ satisfies Anderson localization if $$|\unicode[STIX]{x1D706}|>e^{2\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$$ . In this paper, we develop a method to treat simultaneous frequency and phase resonances and obtain that, for $$2\unicode[STIX]{x1D703}\in \unicode[STIX]{x1D6FC}\mathbb{Z}+\mathbb{Z}$$ , $$H_{\unicode[STIX]{x1D706},\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D703}}$$ satisfies Anderson localization if $$|\unicode[STIX]{x1D706}|>e^{3\unicode[STIX]{x1D6FD}(\unicode[STIX]{x1D6FC})}$$ . 

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4. THE DYNAMICAL MORDELL–LANG CONJECTURE FOR ENDOMORPHISMS OF SEMIABELIAN VARIETIES DEFINED OVER FIELDS OF POSITIVE CHARACTERISTIC

   https://doi.org/10.1017/S1474748019000318  

   Corvaja, Pietro; Ghioca, Dragos; Scanlon, Thomas; Zannier, Umberto (March 2021, Journal of the Institute of Mathematics of Jussieu)

   null (Ed.)

   Abstract Let $$K$$ be an algebraically closed field of prime characteristic $$p$$ , let $$X$$ be a semiabelian variety defined over a finite subfield of $$K$$ , let $$\unicode[STIX]{x1D6F7}:X\longrightarrow X$$ be a regular self-map defined over $$K$$ , let $$V\subset X$$ be a subvariety defined over $$K$$ , and let $$\unicode[STIX]{x1D6FC}\in X(K)$$ . The dynamical Mordell–Lang conjecture in characteristic $$p$$ predicts that the set $$S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$$ is a union of finitely many arithmetic progressions, along with finitely many $$p$$ -sets, which are sets of the form $$\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$$ for some $$m\in \mathbb{N}$$ , some rational numbers $$c_{i}$$ and some non-negative integers $$k_{i}$$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $$X$$ is an algebraic torus, we can prove the conjecture in two cases: either when $$\dim (V)\leqslant 2$$ , or when no iterate of $$\unicode[STIX]{x1D6F7}$$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $$X$$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction. 

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5. POINTWISE CONVERGENCE OF SCHRÖDINGER SOLUTIONS AND MULTILINEAR REFINED STRICHARTZ ESTIMATES

   https://doi.org/10.1017/fms.2018.11  

   DU, XIUMIN; GUTH, LARRY; LI, XIAOCHUN; ZHANG, RUIXIANG (January 2018, Forum of Mathematics, Sigma)

   We obtain partial improvement toward the pointwise convergence problem of Schrödinger solutions, in the general setting of fractal measure. In particular, we show that, for $$n\geqslant 3$$ , $$\lim _{t\rightarrow 0}e^{it\unicode[STIX]{x1D6E5}}f(x)$$ $=f(x)$ almost everywhere with respect to Lebesgue measure for all $$f\in H^{s}(\mathbb{R}^{n})$$ provided that $s>(n+1)/2(n+2)$ . The proof uses linear refined Strichartz estimates. We also prove a multilinear refined Strichartz using decoupling and multilinear Kakeya. 

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