 Award ID(s):
 2054068
 Publication Date:
 NSFPAR ID:
 10320829
 Journal Name:
 International Mathematics Research Notices
 ISSN:
 10737928
 Sponsoring Org:
 National Science Foundation
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Abstract Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$ , the edit distance function $\textrm{ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge additions plus edge deletions applied to a graph of edge density p sufficient to ensure that the resulting graph satisfies $\mathcal{H}$ . The edit distance function is directly related to other wellstudied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$ chromatic number of a random graph. Let $\mathcal{H}$ be the property of forbidding an Erdős–Rényi random graph $F\sim \mathbb{G}(n_0,p_0)$ , and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [11/\varphi,1/\varphi]$ , then a.a.s. as $n_0\to\infty$ , \begin{align*} {\textrm{ed}}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{\log(1p_0)}, \frac{1p}{\log p_0} \right\}. \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$ . A primary tool in the proof is the categorization of p core coloured regularity graphs in the range $p\in[11/\varphi,1/\varphi]$ . Such coloured regularity graphs must have the property that the nongrey edges form vertexdisjoint cliques.

Abstract 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 semianalyticmore »

Abstract 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.

Abstract Given $n$ general points $p_1, p_2, \ldots , p_n \in{\mathbb{P}}^r$ it is natural to ask whether there is a curve of given degree $d$ and genus $g$ passing through them; by counting dimensions a natural conjecture is that such a curve exists if and only if $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d  (r  3)(g  1)}{r  1}\right\rfloor.\end{equation*}$$The case of curves with nonspecial hyperplane section was recently studied in [2], where the above conjecture was shown to hold with exactly three exceptions. In this paper, we prove a “boundederror analog” for special linear series on general curves; more precisely we show that existence of such a curve subject to the stronger inequality $$\begin{equation*}n \leq \left\lfloor \frac{(r + 1)d  (r  3)(g  1)}{r  1}\right\rfloor  3.\end{equation*}$$Note that the $3$ cannot be replaced with $2$ without introducing exceptions (as a canonical curve in ${\mathbb{P}}^3$ can only pass through nine general points, while a naive dimension count predicts twelve). We also use the same technique to prove that the twist of the normal bundle $N_C(1)$ satisfies interpolation for curves whose degree is sufficiently large relative to their genus, and deduce from this that the number of generalmore »

null (Ed.)Abstract We consider the inequality $f \geqslant f\star f$ for real functions in $L^1({\mathbb{R}}^d)$ where $f\star f$ denotes the convolution of $f$ with itself. We show that all such functions $f$ are nonnegative, which is not the case for the same inequality in $L^p$ for any $1 < p \leqslant 2$, for which the convolution is defined. We also show that all solutions in $L^1({\mathbb{R}}^d)$ satisfy $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x \leqslant \tfrac 12$. Moreover, if $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$, then $f$ must decay fairly slowly: $\int _{{\mathbb{R}}^{\textrm{d}}}x f(x)\ \textrm{d}x = \infty $, and this is sharp since for all $r< 1$, there are solutions with $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x = \tfrac 12$ and $\int _{{\mathbb{R}}^{\textrm{d}}}x^r f(x)\ \textrm{d}x <\infty $. However, if $\int _{{\mathbb{R}}^{\textrm{d}}}f(x)\ \textrm{d}x =: a < \tfrac 12$, the decay at infinity can be much more rapid: we show that for all $a<\tfrac 12$, there are solutions such that for some $\varepsilon>0$, $\int _{{\mathbb{R}}^{\textrm{d}}}e^{\varepsilon x}f(x)\ \textrm{d}x < \infty $.