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 $a_f(p) = 2p^{(k1)/2}\cos \theta_p$. 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 Lfunctions 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 squarefree 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: $$ \left \frac{\pi_{f,I}(x)}{\pi(x)}  \mu_{\textrm{ST}}(I)\right \leq 58.1\frac{\log((k1)N \log{x})}{\sqrt{\log{x}}} \qquad \text{for} \quad x \geq 3. $$ As an application, we give an explicit bound for the number of primes up to x that violate the Atkin–Serre conjecture for f.
Given a sequence $\{Z_d\}_{d\in \mathbb{N}}$ of smooth and compact hypersurfaces in ${\mathbb{R}}^{n1}$, 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 $(D, Z(p_m)\cap D)\sim ({\mathbb{R}}^{n1}, Z_{d_m}),$ i.e. the zero set of $p_m$ in $D$ is isotopic to $Z_{d_m}$ in ${\mathbb{R}}^{n1}$. 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 $0 \leq k \leq n2$ and every sequence of natural numbers $a=\{a_d\}_{d\in \mathbb{N}}$ there is a regular, compact semianalytic more »
 Publication Date:
 NSFPAR ID:
 10120351
 Journal Name:
 The Quarterly Journal of Mathematics
 Volume:
 70
 Issue:
 4
 Page Range or eLocationID:
 p. 13971409
 ISSN:
 00335606
 Publisher:
 Oxford University Press
 Sponsoring Org:
 National Science Foundation
More Like this

Abstract 
Let $p:\mathbb{C}\rightarrow \mathbb{C}$ be a polynomial. The Gauss–Lucas theorem states that its critical points, $p^{\prime }(z)=0$ , are contained in the convex hull of its roots. We prove a stability version whose simplest form is as follows: suppose that $p$ has $n+m$ roots, where $n$ are inside the unit disk, $$\begin{eqnarray}\max _{1\leq i\leq n}a_{i}\leq 1~\text{and}~m~\text{are outside}~\min _{n+1\leq i\leq n+m}a_{i}\geq d>1+\frac{2m}{n};\end{eqnarray}$$ then $p^{\prime }$ has $n1$ roots inside the unit disk and $m$ roots at distance at least $(dnm)/(n+m)>1$ from the origin and the involved constants are sharp. We also discuss a pairing result: in the setting above, for $n$ sufficiently large, each of the $m$ roots has a critical point at distance ${\sim}n^{1}$ .

Abstract Let Γ be a Schottky semigroup in {\mathrm{SL}_{2}(\mathbf{Z})} ,and for {q\in\mathbf{N}} , let {\Gamma(q):=\{\gamma\in\Gamma:\gamma=e~{}(\mathrm{mod}~{}q)\}} be its congruence subsemigroupof level q . Let δ denote the Hausdorff dimension of the limit set of Γ.We prove the following uniform congruence counting theoremwith respect to the family of Euclidean norm balls {B_{R}} in {M_{2}(\mathbf{R})} of radius R :for all positive integer q with no small prime factors, \#(\Gamma(q)\cap B_{R})=c_{\Gamma}\frac{R^{2\delta}}{\#(\mathrm{SL}_{2}(%\mathbf{Z}/q\mathbf{Z}))}+O(q^{C}R^{2\delta\epsilon}) as {R\to\infty} for some {c_{\Gamma}>0,C>0,\epsilon>0} which are independent of q .Our technique also applies to give a similar counting result for the continued fractions semigroup of {\mathrm{SL}_{2}(\mathbf{Z})} ,which arises in the study of Zaremba’s conjecture on continued fractions.

Abstract It has been recently established in David and Mayboroda (Approximation of green functions and domains with uniformly rectifiable boundaries of all dimensions.
arXiv:2010.09793 ) that on uniformly rectifiable sets the Green function is almost affine in the weak sense, and moreover, in some scenarios such Green function estimates are equivalent to the uniform rectifiability of a set. The present paper tackles a strong analogue of these results, starting with the “flagship degenerate operators on sets with lower dimensional boundaries. We consider the elliptic operators associated to a domain$$L_{\beta ,\gamma } = {\text {div}}D^{d+1+\gamma n} \nabla $$ ${L}_{\beta ,\gamma}=\text{div}{D}^{d+1+\gamma n}\nabla $ with a uniformly rectifiable boundary$$\Omega \subset {\mathbb {R}}^n$$ $\Omega \subset {R}^{n}$ of dimension$$\Gamma $$ $\Gamma $ , the now usual distance to the boundary$$d < n1$$ $d<n1$ given by$$D = D_\beta $$ $D={D}_{\beta}$ for$$D_\beta (X)^{\beta } = \int _{\Gamma } Xy^{d\beta } d\sigma (y)$$ ${D}_{\beta}{\left(X\right)}^{\beta}={\int}_{\Gamma}{Xy}^{d\beta}d\sigma \left(y\right)$ , where$$X \in \Omega $$ $X\in \Omega $ and$$\beta >0$$ $\beta >0$ . In this paper we show that the Green function$$\gamma \in (1,1)$$ $\gamma \in (1,1)$G for , with pole at infinity, is well approximated by multiples of$$L_{\beta ,\gamma }$$ ${L}_{\beta ,\gamma}$ , in the sense that the function$$D^{1\gamma }$$ ${D}^{1\gamma}$ satisfies a Carleson measure estimate on$$\big  D\nabla \big (\ln \big ( \frac{G}{D^{1\gamma }} \big )\big )\big ^2$$ $D\nabla (ln(\frac{G}{{D}^{1\gamma}})){}^{2}$ . We underline that the strong and the weak results are different in nature and, of course, at the levelmore »$$\Omega $$ $\Omega $ 
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 nonnegative continuous functions on $K$ and $L$, respectively, so that $\g\_\infty =g(0)=1.$ Then $$\begin{align*} & \frac{\int_Kf}{\left(\int_L g\right)^{\frac{nk}n}K^{\frac kn}} \le \frac n{nk} \left(d_{\textrm{ovr}}(K,\mathcal{B}\mathcal{P}_k^n)\right)^k \max_{H} \frac{\int_{K\cap H} f}{\int_{L\cap H} g}, \end{align*}$$where $K$ stands for volume of proper dimension, $C$ is an absolute constant, the maximum is taken over all $(nk)$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.