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1. 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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2. 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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3. LINEAR AND QUADRATIC UNIFORMITY OF THE MÖBIUS FUNCTION OVER

   https://doi.org/10.1112/S0025579319000032  

   Bienvenu, Pierre-Yves; Lê, Thái Hoàng (January 2019, Mathematika)

   We examine correlations of the Möbius function over $$\mathbb{F}_{q}[t]$$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg f0$$ if $$Q$$ is linear and $$O(q^{-n^{c}})$$ for some absolute constant $c>0$ if $$Q$$ is quadratic. The latter bound may be reduced to $$O(q^{-c^{\prime }n})$$ for some $$c^{\prime }>0$$ when $Q(f)$ is a linear form in the coefficients of $$f^{2}$$ , that is, a Hankel quadratic form, whereas, for general quadratic forms, it relies on a bilinear version of the additive-combinatorial Bogolyubov theorem. 

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4. 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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5. The Kodaira dimension of complex hyperbolic manifolds with cusps

   https://doi.org/10.1112/S0010437X1700762X  

   Bakker, Benjamin; Tsimerman, Jacob (March 2018, Compositio Mathematica)

   We prove a bound relating the volume of a curve near a cusp in a complex ball quotient $$X=\mathbb{B}/\unicode[STIX]{x1D6E4}$$ to its multiplicity at the cusp. There are a number of consequences: we show that for an $$n$$ -dimensional toroidal compactification $$\overline{X}$$ with boundary $$D$$ , $$K_{\overline{X}}+(1-\unicode[STIX]{x1D706})D$$ is ample for $$\unicode[STIX]{x1D706}\in (0,(n+1)/2\unicode[STIX]{x1D70B})$$ , and in particular that $$K_{\overline{X}}$$ is ample for $$n\geqslant 6$$ . By an independent algebraic argument, we prove that every ball quotient of dimension $$n\geqslant 4$$ is of general type, and conclude that the phenomenon famously exhibited by Hirzebruch in dimension 2 does not occur in higher dimensions. Finally, we investigate the applications to the problem of bounding the number of cusps and to the Green–Griffiths conjecture. 

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