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1. A COMPACTNESS PRINCIPLE FOR MAXIMISING SMOOTH FUNCTIONS OVER TOROIDAL GEODESICS

   https://doi.org/10.1017/S0004972718001636  

   STEINERBERGER, STEFAN (August 2019, Bulletin of the Australian Mathematical Society)

   Let $$f\in C^{2}(\mathbb{T}^{2})$$ have mean value 0 and consider $$\begin{eqnarray}\sup _{\unicode[STIX]{x1D6FE}\,\text{closed geodesic}}\frac{1}{|\unicode[STIX]{x1D6FE}|}\biggl|\int _{\unicode[STIX]{x1D6FE}}f\,d{\mathcal{H}}^{1}\biggr|,\end{eqnarray}$$ where $$\unicode[STIX]{x1D6FE}$$ ranges over all closed geodesics $$\unicode[STIX]{x1D6FE}:\mathbb{S}^{1}\rightarrow \mathbb{T}^{2}$$ and $$|\unicode[STIX]{x1D6FE}|$$ denotes its length. We prove that this supremum is always attained. Moreover, we can bound the length of the geodesic $$\unicode[STIX]{x1D6FE}$$ attaining the supremum in terms of the smoothness of the function: for all $$s\geq 2$$ , $$\begin{eqnarray}|\unicode[STIX]{x1D6FE}|^{s}{\lesssim}_{s}\biggl(\max _{|\unicode[STIX]{x1D6FC}|=s}\Vert \unicode[STIX]{x2202}_{\unicode[STIX]{x1D6FC}}f\Vert _{L^{1}(\mathbb{T}^{2})}\biggr)\Vert \unicode[STIX]{x1D6FB}f\Vert _{L^{2}}\Vert f\Vert _{L^{2}}^{-2}.\end{eqnarray}$$ 

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2. On the computability of rotation sets and their entropies

   https://doi.org/10.1017/etds.2018.45  

   BURR, MICHAEL A.; SCHMOLL, MARTIN; WOLF, CHRISTIAN (February 2020, Ergodic Theory and Dynamical Systems)

   null (Ed.)

   Let $$f:X\rightarrow X$$ be a continuous dynamical system on a compact metric space $$X$$ and let $$\unicode[STIX]{x1D6F7}:X\rightarrow \mathbb{R}^{m}$$ be an $$m$$ -dimensional continuous potential. The (generalized) rotation set $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is defined as the set of all $$\unicode[STIX]{x1D707}$$ -integrals of $$\unicode[STIX]{x1D6F7}$$ , where $$\unicode[STIX]{x1D707}$$ runs over all invariant probability measures. Analogous to the classical topological entropy, one can associate the localized entropy $$\unicode[STIX]{x210B}(w)$$ to each $$w\in \text{Rot}(\unicode[STIX]{x1D6F7})$$ . In this paper, we study the computability of rotation sets and localized entropy functions by deriving conditions that imply their computability. Then we apply our results to study the case where $$f$$ is a subshift of finite type. We prove that $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ is computable and that $$\unicode[STIX]{x210B}(w)$$ is computable in the interior of the rotation set. Finally, we construct an explicit example that shows that, in general, $$\unicode[STIX]{x210B}$$ is not continuous on the boundary of the rotation set when considered as a function of $$\unicode[STIX]{x1D6F7}$$ and $$w$$ . In particular, $$\unicode[STIX]{x210B}$$ is, in general, not computable at the boundary of $$\text{Rot}(\unicode[STIX]{x1D6F7})$$ . 

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3. 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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4. 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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5. SOME REFINED RESULTS ON THE MIXED LITTLEWOOD CONJECTURE FOR PSEUDO-ABSOLUTE VALUES

   https://doi.org/10.1017/s1446788718000198  

   LIU, WENCAI (August 2019, Journal of the Australian Mathematical Society)

   In this paper, we study the mixed Littlewood conjecture with pseudo-absolute values. For any pseudo-absolute-value sequence $${\mathcal{D}}$$ , we obtain a sharp criterion such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ for a certain one-parameter family of $$\unicode[STIX]{x1D713}$$ . Also, under a minor condition on pseudo-absolute-value sequences $${\mathcal{D}}_{1},{\mathcal{D}}_{2},\ldots ,{\mathcal{D}}_{k}$$ , we obtain a sharp criterion on a general sequence $$\unicode[STIX]{x1D713}(n)$$ such that for almost every $$\unicode[STIX]{x1D6FC}$$ the inequality $$\begin{eqnarray}|n|_{{\mathcal{D}}_{1}}|n|_{{\mathcal{D}}_{2}}\cdots |n|_{{\mathcal{D}}_{k}}|n\unicode[STIX]{x1D6FC}-p|\leq \unicode[STIX]{x1D713}(n)\end{eqnarray}$$ has infinitely many coprime solutions $$(n,p)\in \mathbb{N}\times \mathbb{Z}$$ . 

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