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1. Marcinkiewicz Averages of Smooth Orthogonal Projections on Sphere

   https://doi.org/10.1007/s00041-022-09966-y  

   Bownik, Marcin; Dziedziul, Karol; Kamont, Anna (October 2022, Journal of Fourier Analysis and Applications)

   Abstract We construct a single smooth orthogonal projection with desired localization whose average under a group action yields the decomposition of the identity operator. For any full rank lattice $$\Gamma \subset \mathbb {R}^d$$ Γ ⊂ R d , a smooth projection is localized in a neighborhood of an arbitrary precompact fundamental domain $$\mathbb {R}^d/\Gamma $$ R d / Γ . We also show the existence of a highly localized smooth orthogonal projection, whose Marcinkiewicz average under the action of SO ( d ), is a multiple of the identity on $$L^2(\mathbb {S}^{d-1})$$ L 2 ( S d - 1 ) . As an application we construct highly localized continuous Parseval frames on the sphere. 

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2. Zeroes of polynomials on definable hypersurfaces: pathologies exist, but they are rare

   https://doi.org/10.1093/qmath/haz022  

   Basu, Saugata; Lerario, Antonio; Natarajan, Abhiram (October 2019, The Quarterly Journal of Mathematics)

   Abstract Given a sequence $$\{Z_d\}_{d\in \mathbb{N}}$$ of smooth and compact hypersurfaces in $${\mathbb{R}}^{n-1}$$, we prove that (up to extracting subsequences) there exists a regular definable hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$ such that each manifold $$Z_d$$ is diffeomorphic to a component of the zero set on $$\Gamma$$ of some polynomial of degree $$d$$. (This is in sharp contrast with the case when $$\Gamma$$ is semialgebraic, where for example the homological complexity of the zero set of a polynomial $$p$$ on $$\Gamma$$ is bounded by a polynomial in $$\deg (p)$$.) More precisely, given the above sequence of hypersurfaces, we construct a regular, compact, semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^{n}$$ containing a subset $$D$$ homeomorphic to a disk, and a family of polynomials $$\{p_m\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that $$(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n-1}, Z_{d_m}),$$ i.e. the zero set of $$p_m$$ in $$D$$ is isotopic to $$Z_{d_m}$$ in $${\mathbb{R}}^{n-1}$$. This says that, up to extracting subsequences, the intersection of $$\Gamma$$ with a hypersurface of degree $$d$$ can be as complicated as we want. We call these ‘pathological examples’. In particular, we show that for every $$0 \leq k \leq n-2$$ and every sequence of natural numbers $$a=\{a_d\}_{d\in \mathbb{N}}$$ there is a regular, compact semianalytic hypersurface $$\Gamma \subset {\mathbb{R}}\textrm{P}^n$$, a subsequence $$\{a_{d_m}\}_{m\in \mathbb{N}}$$ and homogeneous polynomials $$\{p_{m}\}_{m\in \mathbb{N}}$$ of degree $$\deg (p_m)=d_m$$ such that (0.1)$$\begin{equation}b_k(\Gamma\cap Z(p_m))\geq a_{d_m}.\end{equation}$$ (Here $$b_k$$ denotes the $$k$$th Betti number.) This generalizes a result of Gwoździewicz et al. [13]. On the other hand, for a given definable $$\Gamma$$ we show that the Fubini–Study measure, in the Gaussian probability space of polynomials of degree $$d$$, of the set $$\Sigma _{d_m,a, \Gamma }$$ of polynomials verifying (0.1) is positive, but there exists a constant $$c_\Gamma$$ such that $$\begin{equation*}0<{\mathbb{P}}(\Sigma_{d_m, a, \Gamma})\leq \frac{c_{\Gamma} d_m^{\frac{n-1}{2}}}{a_{d_m}}.\end{equation*}$$ This shows that the set of ‘pathological examples’ has ‘small’ measure (the faster $$a$$ grows, the smaller the measure and pathologies are therefore rare). In fact we show that given $$\Gamma$$, for most polynomials a Bézout-type bound holds for the intersection $$\Gamma \cap Z(p)$$: for every $$0\leq k\leq n-2$$ and $t>0$: $$\begin{equation*}{\mathbb{P}}\left(\{b_k(\Gamma\cap Z(p))\geq t d^{n-1} \}\right)\leq \frac{c_\Gamma}{td^{\frac{n-1}{2}}}.\end{equation*}$$ 

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3. Compactness estimates for minimizers of the Alt-Phillips functional of negative exponents

   https://doi.org/10.1515/ans-2022-0055  

   De Silva, Daniela; Savin, Ovidiu (January 2023, Advanced Nonlinear Studies)

   Abstract We investigate the rigidity of global minimizers u ≥ 0 u\ge 0 of the Alt-Phillips functional involving negative power potentials ∫ Ω ( ∣ ∇ u ∣ 2 + u − γ χ { u > 0 } ) d x , γ ∈ ( 0 , 2 ) , \mathop{\int }\limits_{\Omega }(| \nabla u{| }^{2}+{u}^{-\gamma }{\chi }_{\left\{u\gt 0\right\}}){\rm{d}}x,\hspace{1.0em}\gamma \in \left(0,2), when the exponent γ \gamma is close to the extremes of the admissible values. In particular, we show that global minimizers in R n {{\mathbb{R}}}^{n} are one-dimensional if γ \gamma is close to 2 and n ≤ 7 n\le 7 , or if γ \gamma is close to 0 and n ≤ 4 n\le 4 . 

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4. Maximal Polarization for Periodic Configurations on the Real Line

   https://doi.org/10.1093/imrn/rnae003  

   Faulhuber, Markus; Steinerberger, Stefan (February 2024, International Mathematics Research Notices)

   Abstract We prove that among all 1-periodic configurations $$\Gamma $$ of points on the real line $$\mathbb{R}$$ the quantities $$\min _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma )^{2}}$$ and $$\max _{x \in \mathbb{R}} \sum _{\gamma \in \Gamma } e^{- \pi \alpha (x - \gamma )^{2}}$$ are maximized and minimized, respectively, if and only if the points are equispaced and whenever the number of points $$n$$ per period is sufficiently large (depending on $$\alpha $$). This solves the polarization problem for periodic configurations with a Gaussian weight on $$\mathbb{R}$$ for large $$n$$. The first result is shown using Fourier series. The second result follows from the work of Cohn and Kumar on universal optimality and holds for all $$n$$ (independent of $$\alpha $$). 

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5. COUNTABLE LENGTH EVERYWHERE CLUB UNIFORMIZATION

   https://doi.org/10.1017/jsl.2022.78  

   CHAN, WILLIAM; JACKSON, STEPHEN; TRANG, NAM (November 2022, The Journal of Symbolic Logic)

   Abstract Assume $$\mathsf {ZF} + \mathsf {AD}$$ and all sets of reals are Suslin. Let $$\Gamma $$ be a pointclass closed under $$\wedge $$ , $$\vee $$ , $$\forall ^{\mathbb {R}}$$ , continuous substitution, and has the scale property. Let $$\kappa = \delta (\Gamma )$$ be the supremum of the length of prewellorderings on $$\mathbb {R}$$ which belong to $$\Delta = \Gamma \cap \check \Gamma $$ . Let $$\mathsf {club}$$ denote the collection of club subsets of $$\kappa $$ . Then the countable length everywhere club uniformization holds for $$\kappa $$ : For every relation $$R \subseteq {}^{<{\omega _1}}\kappa \times \mathsf {club}$$ with the property that for all $$\ell \in {}^{<{\omega _1}}\kappa $$ and clubs $$C \subseteq D \subseteq \kappa $$ , $$R(\ell ,D)$$ implies $$R(\ell ,C)$$ , there is a uniformization function $$\Lambda : \mathrm {dom}(R) \rightarrow \mathsf {club}$$ with the property that for all $$\ell \in \mathrm {dom}(R)$$ , $$R(\ell ,\Lambda (\ell ))$$ . In particular, under these assumptions, for all $$n \in \omega $$ , $$\boldsymbol {\delta }^1_{2n + 1}$$ satisfies the countable length everywhere club uniformization. 

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