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1. 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)

   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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2. An Unconditional Explicit Bound on the Error Term in the Sato–tate Conjecture

   https://doi.org/10.1093/qmath/haac004  

   Hoey, Alexandra ; Iskander, Jonas ; Jin, Steven ; Trejos Suárez, Fernando ( April 2022 , The Quarterly Journal of Mathematics)

   Let $f(z) = \sum_{n=1}^\infty a_f(n)q^n$ be a holomorphic cuspidal newform with even integral weight $k\geq 2$, level N, trivial nebentypus and no complex multiplication. For all primes p, we may define $\theta_p\in [0,\pi]$ such that $a_f(p) = 2p^{(k-1)/2}\cos \theta_p$. The Sato–Tate conjecture states that the angles θp are equidistributed with respect to the probability measure $\mu_{\textrm{ST}}(I) = \frac{2}{\pi}\int_I \sin^2 \theta \; d\theta$, where $I\subseteq [0,\pi]$. Using recent results on the automorphy of symmetric power L-functions due to Newton and Thorne, we explicitly bound the error term in the Sato–Tate conjecture when f corresponds to an elliptic curve over $\mathbb{Q}$ of arbitrary conductor or when f has square-free level. In these cases, if $\pi_{f,I}(x) := \#\{p \leq x : p \nmid N, \theta_p\in I\}$ and $\pi(x) := \# \{p \leq x \}$, we prove the following bound: $$ \left| \frac{\pi_{f,I}(x)}{\pi(x)} - \mu_{\textrm{ST}}(I)\right| \leq 58.1\frac{\log((k-1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3. $$ As an application, we give an explicit bound for the number of primes up to x that violate the Atkin–Serre conjecture for f.

    

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3. Uniqueness and Symmetry for the Mean Field Equation on Arbitrary Flat Tori

   https://doi.org/10.1093/imrn/rnaa109  

   Gu, Guangze ; Gui, Changfeng ; Hu, Yeyao ; Li, Qinfeng ( May 2020 , International Mathematics Research Notices)

   We study the following mean field equation on a flat torus $T:=\mathbb{C}/(\mathbb{Z}+\mathbb{Z}\tau )$: $$\begin{equation*} \varDelta u + \rho \left(\frac{e^{u}}{\int_{T}e^u}-\frac{1}{|T|}\right)=0, \end{equation*}$$where $ \tau \in \mathbb{C}, \mbox{Im}\ \tau>0$, and $|T|$ denotes the total area of the torus. We first prove that the solutions are evenly symmetric about any critical point of $u$ provided that $\rho \leq 8\pi $. Based on this crucial symmetry result, we are able to establish further the uniqueness of the solution if $\rho \leq \min{\{8\pi ,\lambda _1(T)|T|\}}$. Furthermore, we also classify all one-dimensional solutions by showing that the level sets must be closed geodesics.

    

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4. On lower bounds for Erdős-Szekeres products

   https://doi.org/10.1090/proc/15503  

   Billsborough, C. ; Freedman, M. ; Hart, S. ; Kowalsky, G. ; Lubinsky, D. ; Pomeranz, A. ; Sammel, A. ( October 2021 , Proceedings of the American Mathematical Society)

   Let { s j } j = 1 n \left \{ s_{j}\right \} _{j=1}^{n} be positive integers. We show that for any 1 ≤ L ≤ n , 1\leq L\leq n, ‖ ∏ j = 1 n ( 1 − z s j ) ‖ L ∞ ( | z | = 1 ) ≥ exp ⁡ ( 1 2 e L ( s 1 s 2 … s L ) 1 / L ) . \begin{equation*} \left \Vert \prod _{j=1}^{n}\left ( 1-z^{s_{j}}\right ) \right \Vert _{L_{\infty }\left ( \left \vert z\right \vert =1\right ) }\geq \exp \left ( \frac {1}{2e}\frac {L}{\left ( s_{1}s_{2}\ldots s_{L}\right ) ^{1/L}}\right ) . \end{equation*} In particular, this gives geometric growth if a positive proportion of the { s j } \left \{ s_{j}\right \} are bounded. We also show that when the { s j } \left \{ s_{j}\right \} grow regularly and faster than j ( log ⁡ j ) 2 + ε j\left ( \log j\right ) ^{2+\varepsilon } , some ε > 0 \varepsilon >0 , then the norms grow faster than exp ⁡ ( ( log ⁡ n ) 1 + δ ) \exp \left ( \left ( \log n\right ) ^{1+\delta }\right ) for some δ > 0 \delta >0 . 

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5. Inequalities for the Radon Transform on Convex Sets

   https://doi.org/10.1093/imrn/rnab122  

   Giannopoulos, Apostolos ; Koldobsky, Alexander ; Zvavitch, Artem ( May 2021 , International Mathematics Research Notices)

   Abstract We prove an inequality that unifies previous works of the authors on the properties of the Radon transform on convex bodies including an extension of the Busemann–Petty problem and a slicing inequality for arbitrary functions. Let $K$ and $L$ be star bodies in ${\mathbb R}^n,$ let $0<k<n$ be an integer, and let $f,g$ be non-negative continuous functions on $K$ and $L$, respectively, so that $\|g\|_\infty =g(0)=1.$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{n-k}n}|K|^{\frac kn}} \le \frac n{n-k} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $|K|$ stands for volume of proper dimension, $C$ is an absolute constant, the maximum is taken over all $(n-k)$-dimensional subspaces of ${\mathbb R}^n,$ and $d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)$ is the outer volume ratio distance from $K$ to the class of generalized $k$-intersection bodies in ${\mathbb R}^n.$ Another consequence of this result is a mean value inequality for the Radon transform. We also obtain a generalization of the isomorphic version of the Shephard problem. 

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