Recent zbMATH articles in MSC 11Hhttps://zbmath.org/atom/cc/11H2021-05-28T16:06:00+00:00WerkzeugOn the sup-norm of \(\mathrm{SL}_3\) Hecke-Maass cusp forms.https://zbmath.org/1459.111192021-05-28T16:06:00+00:00"Holowinsky, Roman"https://zbmath.org/authors/?q=ai:holowinsky.roman"Nowland, Kevin"https://zbmath.org/authors/?q=ai:nowland.kevin"Ricotta, Guillaume"https://zbmath.org/authors/?q=ai:ricotta.guillaume"Royer, Emmanuel"https://zbmath.org/authors/?q=ai:royer.emmanuelLet \(X:=\mathrm{SL}_3(\mathbb{Z}) \setminus \mathrm{SL}_3(\mathbb{R}) / \mathrm{SO}_3(\mathbb{R})\) be non-compact Riemannian symmetric space of dimension \(5\) and rank \(2\).
In the paper under review, the authors give a proof of a non-trivial explicit quantitative bound in the eigenvalue aspect for the sup-norm of a \(\mathrm{SL}_3(\mathbb{Z})\) Hecke-Maass cusp form at a generic point \(z\) in a fixed compact subset of \(X\).
More precisely, they prove the following result: Let \(\Phi\) be an \(L^2\)-normalized and tempered \(\mathrm{SL}_3(\mathbb{Z})\) Hecke-Maass cusp form on \(X\) with Laplace eigenvalue \(\lambda\) and type \((v_1, v_2)\) in \(i \mathbb{R}^2\) satisfying \(|v_1 -v_2| \ll 1\). Let \(C\) be a fixed compact in \(X\). Then one has
\[ \| \Phi \mid_{C} \|_\infty \ll_{C,\epsilon}\lambda^{(5-2)/4-1/76+\epsilon} . \]
The proof is based on some calculations by Rankin-Selberg theory and the Cauchy-Schwarz inequality.
For the entire collection see [Zbl 1437.11002].
Reviewer: Ilker Inam (Bilecik)Vulnerable public keys in NTRU cryptosystem.https://zbmath.org/1459.941482021-05-28T16:06:00+00:00"Xu, Liqing"https://zbmath.org/authors/?q=ai:xu.liqing"Chen, Hao"https://zbmath.org/authors/?q=ai:chen.hao.1"Li, Chao"https://zbmath.org/authors/?q=ai:li.chao"Qu, Longjiang"https://zbmath.org/authors/?q=ai:qu.longjiangSummary: In this paper the authors give an efficient bounded distance decoding (BDD for short) algorithm for NTRU lattices under some conditions about the modulus number \(q\) and the public key \(\mathfrak{h}\). They then use this algorithm to give plain-text recovery attack to NTRU \textit{Encrypt} and forgery attack on NTRU \textit{Sign}. In particular the authors figure out a weak domain of public keys such that the recent transcript secure version of NTRU signature scheme NTRUMLS with public keys in this domain can be forged.Some results on random unimodular lattices.https://zbmath.org/1459.111452021-05-28T16:06:00+00:00"Skenderi, Mishel"https://zbmath.org/authors/?q=ai:skenderi.mishelThe author generalizes several results from [\textit{C. A. Rogers}, Acta Math. 94, 249--287 (1955; Zbl 0065.28201); \textit{I. Aliev} and \textit{P. M. Gruber}, Discrete Comput. Geom. 35, No. 3, 429--435 (2006; Zbl 1215.11066); \textit{J. S. Athreya} and \textit{G. A. Margulis}, J. Mod. Dyn. 3, No. 3, 359--378 (2009; Zbl 1184.37007)] by showing that there exists a real number greater than or equal to 1, \(\omega_n\), such that for any Borel subset \(A\) of \(n\) dimensional Euclidean space with \(0 < m(A) < \infty\), we have
\[\eta(\{\Lambda \in X_n \mid \dim_{\mathbf{R}}(\mathrm{span}_{\mathbf{R}} (A \cap \Lambda_{pr})) < n-2 \}) \leq\frac{\omega_n}{m(A)}\]
where \(X_n\) is the space of all full rank unimodular lattices in \(\mathbf{R}^n\), \(\eta\) is the Haar probability measure on this space, and \(\Lambda_{pr} = \{ v \in \Lambda \mid \exists\mathbf{Z}\)-basis \(\mathcal{B}\) of \(\Lambda\) for which \(v\in\mathcal{B}\}\). Additionally, independent results by Rogers [loc. cit.] and \textit{W. Schmidt} [Trans. Am. Math. Soc. 95, 516--529 (1960; Zbl 0101.27904)] about primitive lattice points of random lattices to the case of primitive tuples of rank less than \(\frac{n}{2}\) are generalized.
Reviewer: Steven T. Dougherty (Scranton)Simultaneous approximation to values of the exponential function over the adeles.https://zbmath.org/1459.111472021-05-28T16:06:00+00:00"Roy, Damien"https://zbmath.org/authors/?q=ai:roy.damienUsing the continued fraction expansion of \(e\) and \(e^{2/m}\) by Euler, Sundman and Hurwitz one may derive very good measures of rational approximations to these numbers, see, e.g., \textit{P. Bundschuh} [Math. Ann. 192, 229--242 (1971; Zbl 0206.05803)].
The author shows that Hermite's approximations to values of the exponential function at given algebraic numbers are nearly optimal when considered from an adelic perspective. The main result is a far reaching simultaneous generalization of the following theorem:
Let \(K \subset \mathbb C\) be the field \(\mathbb Q\) or a quadratic imaginary extension of \(\mathbb Q\), and let \(\alpha\) be a non-zero element of \(K\) such that \(|\alpha|_v \ge p^{-1/(p-1)}\) for each prime number \(p\) and each place \(v\) of \(K\) with \(v | p\). Then, for any \(x, y \in\mathcal{O}_K\) with \(|x| > 1\), we have
\[
|x|\cdot |xe^{\alpha}-y| \ge c(\log |x|)^{-2g-1},
\]
where \(g\) stands for the number of places \(v\) of \(K\) with \(v | \infty\) or \(|\alpha|_v\ne 1\), and where \(c \geq 0\) is a constant depending only on \(\alpha\) and \(K\).
Reviewer: István Gaál (Debrecen)