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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A COMPACTNESS PRINCIPLE FOR MAXIMISING SMOOTH FUNCTIONS OVER TOROIDAL GEODESICS
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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- Award ID(s):
- 1763179
- NSF-PAR ID:
- 10133829
- Date Published:
- Journal Name:
- Bulletin of the Australian Mathematical Society
- Volume:
- 100
- Issue:
- 1
- ISSN:
- 0004-9727
- Page Range / eLocation ID:
- 148 to 154
- Format(s):
- Medium: X
- Sponsoring Org:
- National Science Foundation
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We give an integrability criterion on a real-valued non-increasing function $\unicode[STIX]{x1D713}$ guaranteeing that for almost all (or almost no) pairs $(A,\mathbf{b})$ , where $A$ is a real $m\times n$ matrix and $\mathbf{b}\in \mathbb{R}^{m}$ , the system $$\begin{eqnarray}\Vert A\mathbf{q}+\mathbf{b}-\mathbf{p}\Vert ^{m}<\unicode[STIX]{x1D713}(T),\quad \Vert \mathbf{q}\Vert ^{n}We examine correlations of the Möbius function over $\mathbb{F}_{q}[t]$ with linear or quadratic phases, that is, averages of the form 1 $$\begin{eqnarray}\frac{1}{q^{n}}\mathop{\sum }_{\deg fmore » « lessFor each $t\in \mathbb{R}$ , we define the entire function $$\begin{eqnarray}H_{t}(z):=\int _{0}^{\infty }e^{tu^{2}}\unicode[STIX]{x1D6F7}(u)\cos (zu)\,du,\end{eqnarray}$$ where $\unicode[STIX]{x1D6F7}$ is the super-exponentially decaying function $$\begin{eqnarray}\unicode[STIX]{x1D6F7}(u):=\mathop{\sum }_{n=1}^{\infty }(2\unicode[STIX]{x1D70B}^{2}n^{4}e^{9u}-3\unicode[STIX]{x1D70B}n^{2}e^{5u})\exp (-\unicode[STIX]{x1D70B}n^{2}e^{4u}).\end{eqnarray}$$ Newman showed that there exists a finite constant $\unicode[STIX]{x1D6EC}$ (the de Bruijn–Newman constant ) such that the zeros of $H_{t}$ are all real precisely when $t\geqslant \unicode[STIX]{x1D6EC}$ . The Riemann hypothesis is equivalent to the assertion $\unicode[STIX]{x1D6EC}\leqslant 0$ , and Newman conjectured the complementary bound $\unicode[STIX]{x1D6EC}\geqslant 0$ . In this paper, we establish Newman’s conjecture. The argument proceeds by assuming for contradiction that $\unicode[STIX]{x1D6EC}<0$ and then analyzing the dynamics of zeros of $H_{t}$ (building on the work of Csordas, Smith and Varga) to obtain increasingly strong control on the zeros of $H_{t}$ in the range $\unicode[STIX]{x1D6EC}null (Ed.)Abstract Let $K$ be an algebraically closed field of prime characteristic $p$ , let $X$ be a semiabelian variety defined over a finite subfield of $K$ , let $\unicode[STIX]{x1D6F7}:X\longrightarrow X$ be a regular self-map defined over $K$ , let $V\subset X$ be a subvariety defined over $K$ , and let $\unicode[STIX]{x1D6FC}\in X(K)$ . The dynamical Mordell–Lang conjecture in characteristic $p$ predicts that the set $S=\{n\in \mathbb{N}:\unicode[STIX]{x1D6F7}^{n}(\unicode[STIX]{x1D6FC})\in V\}$ is a union of finitely many arithmetic progressions, along with finitely many $p$ -sets, which are sets of the form $\{\sum _{i=1}^{m}c_{i}p^{k_{i}n_{i}}:n_{i}\in \mathbb{N}\}$ for some $m\in \mathbb{N}$ , some rational numbers $c_{i}$ and some non-negative integers $k_{i}$ . We prove that this conjecture is equivalent with some difficult diophantine problem in characteristic 0. In the case $X$ is an algebraic torus, we can prove the conjecture in two cases: either when $\dim (V)\leqslant 2$ , or when no iterate of $\unicode[STIX]{x1D6F7}$ is a group endomorphism which induces the action of a power of the Frobenius on a positive dimensional algebraic subgroup of $X$ . We end by proving that Vojta’s conjecture implies the dynamical Mordell–Lang conjecture for tori with no restriction.more » « less
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Accepted Manuscript1.0
- Journal Article:
- https://doi.org/10.1017/S0004972718001636
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