Newforms with rational coefficients

We consider the set of classical newforms with rational coefficients and no complex multiplication. We study the distribution of quadratic twist-classes of these forms with respect to weight k and minimal level N. We conjecture that for each weight \(k \ge 6\), there are only finitely many classes. In large weights, we make this conjecture effective: in weights \(18 \le k \le 24\), all classes have \(N \le 30\); in weights \(26 \le k \le 50\), all classes have \(N \in \{2,6\}\); and in weights \(k \ge 52\), there are no classes at all. We study some of the newforms appearing on our conjecturally complete list in more detail, especially in the cases \(N=2\), 3, 4, 6, and 8, where formulas can be kept nearly as simple as those for the classical case \(N=1\).     

Isomorphic copies of $$l^\infty $$ in Cesàro–Orlicz function spaces

We characterize Cesàro–Orlicz function spaces \(Ces_\varphi \) containing isomorphic copy of \(l^\infty \). We also describe the subspaces \((Ces_\varphi )_a\) of all order continuous elements of \(Ces_\varphi \). Finally, we study the monotonicity structure of the spaces \(Ces_\varphi \) and \((Ces_\varphi )_a\).     

Percolation of random nodal lines

We prove a Russo-Seymour-Welsh percolation theorem for nodal domains and nodal lines associated to a natural infinite dimensional space of real analytic functions on the real plane. More precisely, let \(U\) be a smooth connected bounded open set in \(\mathbf{R}^{2}\) and \(\gamma, \gamma '\) two disjoint arcs of positive length in the boundary of \(U\). We prove that there exists a positive constant \(c\), such that for any positive scale \(s\), with probability at least \(c\) there exists a connected component of the set \(\{x\in \smash{\bar{U}},\ f(sx) > 0\} \) intersecting both \(\gamma \) and \(\gamma '\), where \(f\) is a random analytic function in the Wiener space associated to the real Bargmann-Fock space. For \(s\) large enough, the same conclusion holds for the zero set \(\{x\in \smash{\bar{U}},\ f(sx) = 0\} \). As an important intermediate result, we prove that sign percolation for a general stationary Gaussian field can be made equivalent to a correlated percolation model on a lattice.     

On generalized Ramanujan primes

Let \(\Delta = \sum _{m=0}^\infty q^{(2m+1)^2} \in \mathbf {F}_2[[q]]\) be the reduction mod 2 of the \(\Delta \) series. A modular form of level 1, \(f=\sum _{n\geqslant 0} c(n) \,q^n\), with integer coefficients, is congruent modulo \(2\) to a polynomial in \(\Delta \). Let us set \(W_f(x)=\sum _{n\leqslant x,\ c(n)\text { odd }} 1\), the number of odd Fourier coefficients of \(f\) of index \(\leqslant x\). The order of magnitude of \(W_f(x)\) (for \(x\rightarrow \infty \)) has been determined by Serre in the seventies. Here, we give an asymptotic equivalent for \(W_f(x)\). Let \(p(n)\) be the partition function and \(A_0(x)\) (resp. \(A_1(x)\)) be the number of \(n\leqslant x\) such that \(p(n)\) is even (resp. odd). In the preceding papers, the second-named author has shown that \(A_0(x)\geqslant 0.28 \sqrt{x\;\log \log x}\) for \(x\geqslant 3\) and \(A_1(x)>\frac{4.57 \sqrt{x}}{\log x}\) for \(x\geqslant 7\). Here, it is proved that \(A_0(x)\geqslant 0.069 \sqrt{x}\;\log \log x\) holds for \(x>1\) and that \(A_1(x) \geqslant \frac{0.037 \sqrt{x}}{(\log x)^{7/8}}\) holds for \(x\geqslant 2\). The main tools used to prove these results are the determination of the order of nilpotence of a modular form of level-\(1\) modulo \(2\), and of the structure of the space of those modular forms as a module over the Hecke algebra, which have been given in a recent work of Serre and the second-named author.     

Brownian Paths Homogeneously Distributed in Space: Percolation Phase Transition and Uniqueness of the Unbounded Cluster

We consider a continuum percolation model on \(\mathbb {R}^d\), \(d\ge 1\). For \(t,\lambda \in (0,\infty )\) and \(d\in \{1,2,3\}\), the occupied set is given by the union of independent Brownian paths running up to time t whose initial points form a Poisson point process with intensity \(\lambda >0\). When \(d\ge 4\), the Brownian paths are replaced by Wiener sausages with radius \(r>0\). We establish that, for \(d=1\) and all choices of t, no percolation occurs, whereas for \(d\ge 2\), there is a non-trivial percolation transition in t, provided \(\lambda \) and r are chosen properly. The last statement means that \(\lambda \) has to be chosen to be strictly smaller than the critical percolation parameter for the occupied set at time zero (which is infinite when \(d\in \{2,3\}\), but finite and dependent on r when \(d\ge 4\)). We further show that for all \(d\ge 2\), the unbounded cluster in the supercritical phase is unique. Along the way a finite box criterion for non-percolation in the Boolean model is extended to radius distributions with an exponential tail. This may be of independent interest. The present paper settles the basic properties of the model and should be viewed as a springboard for finer results.     

New nonbinary code bounds based on divisibility arguments

For \(q,n,d \in \mathbb {N}\), let \(A_q(n,d)\) be the maximum size of a code \(C \subseteq [q]^n\) with minimum distance at least d. We give a divisibility argument resulting in the new upper bounds \(A_5(8,6) \le 65\), \(A_4(11,8)\le 60\) and \(A_3(16,11) \le 29\). These in turn imply the new upper bounds \(A_5(9,6) \le 325\), \(A_5(10,6) \le 1625\), \(A_5(11,6) \le 8125\) and \(A_4(12,8) \le 240\). Furthermore, we prove that for \(\mu ,q \in \mathbb {N}\), there is a 1–1-correspondence between symmetric \((\mu ,q)\)-nets (which are certain designs) and codes \(C \subseteq [q]^{\mu q}\) of size \(\mu q^2\) with minimum distance at least \(\mu q - \mu \). We derive the new upper bounds \(A_4(9,6) \le 120\) and \(A_4(10,6) \le 480\) from these ‘symmetric net’ codes.     

Gravitational Field Equations on Fefferman Space–Times

The total space \({\mathfrak M} \approx {\mathbb H}_1 \times S^1\) of the canonical circle bundle over the 3-dimensional Heisenberg group \({\mathbb H}_1\) is a space–time with the Lorentzian metric \(F_{\theta _0}\) (Fefferman’s metric) associated to the canonical Tanaka–Webster flat contact form \(\theta _0\) on \({\mathbb H}_1\). The matter and energy content of \(\mathfrak M\) is described by the energy-momentum tensor \({T}_{\mu \nu }\) (the trace-less Ricci tensor of \(F_{\theta _0}\)) as an effect of the non flat nature of Feferman’s metric \(F_{\theta _0}\). We study the gravitational field equations \(R_{\mu \nu } - (1/2) \, R \, g_{\mu \nu } = {T}_{\mu \nu }\) on \({\mathfrak M}\). We consider the first order perturbation \(g = F_{\theta _0} + \epsilon \, h\), \(\epsilon<< 1\), and linearize the field equations about \(F_{\theta _0}\). We determine a Lorentzian metric g on \({\mathfrak M}\) which solves the linearized field equations corresponding to a diagonal perturbation h.     

Standard Jordan partitions with three parameters

We extend previous work on standard two-parameter Jordan partitions by Barry (Commun Algebra 43:4231–4246, 2015) to three parameters. Let \(J_r\) denote an \(r \times r\) matrix with minimal polynomial \((t-1)^r\) over a field F of characteristic p. For positive integers \(n_1\), \(n_2\), and \(n_3\) satisfying \(n_1 \le n_2 \le n_3\), the Jordan canonical form of the \(n_1 n_2 n_3 \times n_1 n_2 n_3\) matrix \(J_{n_1} \otimes J_{n_2} \otimes J_{n_3}\) has the form \(J_{\lambda _1} \oplus J_{\lambda _2} \oplus \cdots \oplus J_{\lambda _m}\) where \(\lambda _1 \ge \lambda _2 \ge \cdots \ge \lambda _m>0\) and \(\sum _{i=1}^m \lambda _i=n_1 n_2 n_3\). The partition \(\lambda (n_1,n_2,n_3:p)=(\lambda _1, \lambda _2,\ldots , \lambda _m)\) of \(n_1 n_2 n_3\), which depends on \(n_1\), \(n_2\), \(n_3\), and p, will be called a Jordan partition. We will define what we mean by a standard Jordan partition and give necessary and sufficient conditions for its existence.     

Large conformal metrics with prescribed sign-changing Gauss curvature

Let \((M,g)\) be a two dimensional compact Riemannian manifold of genus \(g(M)>1\). Let \(f\) be a smooth function on \(M\) such that

$$\begin{aligned} f \ge 0, \quad f\not \equiv 0, \quad \min _M f = 0. \end{aligned}$$

Let \(p_1,\ldots ,p_n\) be any set of points at which \(f(p_i)=0\) and \(D^2f(p_i)\) is non-singular. We prove that for all sufficiently small \(\lambda >0\) there exists a family of “bubbling” conformal metrics \(g_\lambda =e^{u_\lambda }g\) such that their Gauss curvature is given by the sign-changing function \(K_{g_\lambda }=-f+\lambda ^2\). Moreover, the family \(u_\lambda \) satisfies

$$\begin{aligned} u_\lambda (p_j) = -4\log \lambda -2\log \left( \frac{1}{\sqrt{2}} \log \frac{1}{\lambda }\right) +O(1) \end{aligned}$$

and

$$\begin{aligned} \lambda ^2e^{u_\lambda }\rightharpoonup 8\pi \sum _{i=1}^{n}\delta _{p_i},\quad \text{ as } \lambda \rightarrow 0, \end{aligned}$$

where \(\delta _{p}\) designates Dirac mass at the point \(p\).

     

Elimination of extremal index zeroes from generic paths of closed $$$$-forms

Let \( \alpha \) be a Morse closed \( 1 \)-form of a smooth \( n \)-dimensional manifold \( M \). The zeroes of \( \alpha \) of index \( 0 \) or \( n \) are called centers. It is known that every non-vanishing de Rham cohomology class \( u \) contains a Morse representative without centers. The result of this paper is the one-parameter analogue of the last statement: every generic path \( (\alpha _t)_{ t\in [0,1] }\) of closed \( 1 \)-forms in a fixed class \( u\ne 0 \) such that \( \alpha _0,\alpha _1 \) have no centers, can be modified relatively to its extremities to another such path \( (\beta _t)_{t \in [0,1]} \) having no center at all.     

Cover-free codes and separating system codes

A set \(T\subset {GF(q)}\), \(q=p^h\) is a super-Vandermonde set if \(\sum _{y\in T} y^k=0\) for \(0< k <|T|\). We determine the structure of super-Vandermonde sets of size \(p+1\) (almost small) and size \(q/p-1\) (almost large).     

Partial permutation decoding for binary linear and $$Z_4$$-linear Hadamard codes

In this paper, s-\({\text {PD}}\)-sets of minimum size \(s+1\) for partial permutation decoding for the binary linear Hadamard code \(H_m\) of length \(2^m\), for all \(m\ge 4\) and \(2 \le s \le \lfloor {\frac{2^m}{1+m}}\rfloor -1\), are constructed. Moreover, recursive constructions to obtain s-\({\text {PD}}\)-sets of size \(l\ge s+1\) for \(H_{m+1}\) of length \(2^{m+1}\), from an s-\({\text {PD}}\)-set of the same size for \(H_m\), are also described. These results are generalized to find s-\({\text {PD}}\)-sets for the \({\mathbb {Z}}_4\)-linear Hadamard codes \(H_{\gamma , \delta }\) of length \(2^m\), \(m=\gamma +2\delta -1\), which are binary Hadamard codes (not necessarily linear) obtained as the Gray map image of quaternary linear codes of type \(2^\gamma 4^\delta \). Specifically, s-PD-sets of minimum size \(s+1\) for \(H_{\gamma , \delta }\), for all \(\delta \ge 3\) and \(2\le s \le \lfloor {\frac{2^{2\delta -2}}{\delta }}\rfloor -1\), are constructed and recursive constructions are described.     

CCZ equivalence of power functions

Let \(F\simeq {{\mathrm{GF}}}(p^n)\) be a finite field of characteristic p and \(p_k\) and \(p_\ell \) be power functions on F defined by \(p_k(x)=x^k\) and \(p_\ell (x)=x^\ell \) respectively. We show, that \(p_k\) and \(p_\ell \) are CCZ equivalent, if and only if there exists a positive integer \(0\le a< n\), such that \(\ell \equiv p^a k \pmod {p^n-1}\) or \(k\ell \equiv p^a \pmod {p^n-1}\).     

Improved upper bounds for partial spreads

A partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) is a collection of \((k-1)\)-dimensional subspaces with trivial intersection. So far, the maximum size of a partial \((k-1)\)-spread in \({\text {PG}}(n-1,q)\) was known for the cases \(n\equiv 0\pmod k\), \(n\equiv 1\pmod k\), and \(n\equiv 2\pmod k\) with the additional requirements \(q=2\) and \(k=3\). We completely resolve the case \(n\equiv 2\pmod k\) for the binary case \(q=2\).     

The Sharp Weighted Bound for Multilinear Maximal Functions and Calderón–Zygmund Operators

We investigate the weighted bounds for multilinear maximal functions and Calderón–Zygmund operators from \(L^{p_1}(w_1)\times \cdots \times L^{p_m}(w_m)\) to \(L^{p}(v_{\vec {w}})\), where \(1<p_1,\cdots ,p_m<\infty \) with \(1/{p_1}+\cdots +1/{p_m}=1/p\) and \(\vec {w}\) is a multiple \(A_{\vec {P}}\) weight. We prove the sharp bound for the multilinear maximal function for all such \(p_1,\ldots , p_m\) and prove the sharp bound for \(m\)-linear Calderón–Zymund operators when \(p\ge 1\).     

On discrete universality of the Riemann zeta-function with respect to uniformly distributed shifts

The Voronin universality theorem asserts that a wide class of analytic functions can be approximated by shifts \(\zeta (s+i\tau )\), \(\tau \in \mathbb {R}\), of the Riemann zeta-function. In the paper, we obtain a universality theorem on the approximation of analytic functions by discrete shifts \(\zeta (s+ix_kh)\), \(k\in \mathbb {N}\), \(h>0\), where \(\{x_k\}\subset \mathbb {R}\) is such that the sequence \(\{ax_k\}\) with every real \(a\ne 0\) is uniformly distributed modulo 1, \(1\le x_k\le k\) for all \(k\in \mathbb {N}\) and, for \(1\le k\), \(m\le N\), \(k\ne m\), the inequality \(|x_k-x_m| \ge y^{-1}_N\) holds with \(y_N> 0\) satisfying \(y_Nx_N\ll N\).     

Ramanujan-type congruences for overpartitions modulo 16

Let \(\bar{p}(n)\) denote the number of overpartitions of \(n\). Recently, Fortin–Jacob–Mathieu and Hirschhorn–Sellers independently obtained 2-, 3- and 4-dissections of the generating function for \(\bar{p}(n)\) and derived a number of congruences for \(\bar{p}(n)\) modulo 4, 8 and 64 including \(\bar{p}(8n+7)\equiv 0 \pmod {64}\) for \(n\ge 0\). In this paper, we give a 16-dissection of the generating function for \(\bar{p}(n)\) modulo 16 and show that \(\bar{p}(16n+14)\equiv 0\pmod {16}\) for \(n\ge 0\). Moreover, using the \(2\)-adic expansion of the generating function for \(\bar{p}(n)\) according to Mahlburg, we obtain that \(\bar{p}(\ell ^2n+r\ell )\equiv 0\pmod {16}\), where \(n\ge 0\), \(\ell \equiv -1\pmod {8}\) is an odd prime and \(r\) is a positive integer with \(\ell \not \mid r\). In particular, for \(\ell =7\) and \(n\ge 0\), we get \(\bar{p}(49n+7)\equiv 0\pmod {16}\) and \(\bar{p}(49n+14)\equiv 0\pmod {16}\). We also find four congruence relations: \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n) \pmod {16}\) for \(n\ge 0\), \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {32}\) where \(n\) is not a square of an odd positive integer, \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {64}\) for \(n\not \equiv 1,2,5\pmod {8}\) and \(\bar{p}(4n)\equiv (-1)^n\bar{p}(n)\pmod {128}\) for \(n\equiv 0\pmod {4}\).     

An Erdös–Révész Type Law of the Iterated Logarithm for Order Statistics of a Stationary Gaussian Process

Let \(\{X(t):t\in \mathbb R_+\}\) be a stationary Gaussian process with almost surely (a.s.) continuous sample paths, \(\mathbb E X(t) = 0, \mathbb E X^2(t) = 1\) and correlation function satisfying (i) \(r(t) = 1 - C|t|^{\alpha } + o(|t|^{\alpha })\) as \(t\rightarrow 0\) for some \(0\le \alpha \le 2\) and \(C>0\); (ii) \(\sup _{t\ge s}|r(t)|<1\) for each \(s>0\) and (iii) \(r(t) = O(t^{-\lambda })\) as \(t\rightarrow \infty \) for some \(\lambda >0\). For any \(n\ge 1\), consider n mutually independent copies of X and denote by \(\{X_{r:n}(t):t\ge 0\}\) the rth smallest order statistics process, \(1\le r\le n\). We provide a tractable criterion for assessing whether, for any positive, non-decreasing function \(f, \mathbb P(\mathscr {E}_f)=\mathbb P(X_{r:n}(t) > f(t)\, \text { i.o.})\) equals 0 or 1. Using this criterion we find, for a family of functions \(f_p(t)\) such that \(z_p(t)=\mathbb P(\sup _{s\in [0,1]}X_{r:n}(s)>f_p(t))=O((t\log ^{1-p} t)^{-1})\), that \(\mathbb P(\mathscr {E}_{f_p})= 1_{\{p\ge 0\}}\). Consequently, with \(\xi _p (t) = \sup \{s:0\le s\le t, X_{r:n}(s)\ge f_p(s)\}\), for \(p\ge 0\) we have \(\lim _{t\rightarrow \infty }\xi _p(t)=\infty \) and \(\limsup _{t\rightarrow \infty }(\xi _p(t)-t)=0\) a.s. Complementarily, we prove an Erdös–Révész type law of the iterated logarithm lower bound on \(\xi _p(t)\), namely, that \(\liminf _{t\rightarrow \infty }(\xi _p(t)-t)/h_p(t) = -1\) a.s. for \(p>1\) and \(\liminf _{t\rightarrow \infty }\log (\xi _p(t)/t)/(h_p(t)/t) = -1\) a.s. for \(p\in (0,1]\), where \(h_p(t)=(1/z_p(t))p\log \log t\).     

Strong extensions for q-summing o perators acting in p-convex Banach function s paces for $$1 \le p \le q$$1≤ p≤ q

Let \(1\le p\le q<\infty \) and let X be a p-convex Banach function space over a \(\sigma \)-finite measure \(\mu \). We combine the structure of the spaces \(L^p(\mu )\) and \(L^q(\xi )\) for constructing the new space \(S_{X_p}^{\,q}(\xi )\), where \(\xi \) is a probability Radon measure on a certain compact set associated to X. We show some of its properties, and the relevant fact that every q-summing operator T defined on X can be continuously (strongly) extended to \(S_{X_p}^{\,q}(\xi )\). Our arguments lead to a mixture of the Pietsch and Maurey-Rosenthal factorization theorems, which provided the known (strong) factorizations for q-summing operators through \(L^q\)-spaces when \(1 \le q \le p\). Thus, our result completes the picture, showing what happens in the complementary case \(1\le p\le q\).     

Weighted Harmonic Bloch Spaces on the Ball

We study the family of weighted harmonic Bloch spaces \(b_\alpha , \alpha \in {\mathbb {R}}\), on the unit ball of \({\mathbb {R}}^n\). We provide characterizations in terms of partial and radial derivatives and certain radial differential operators that are more compatible with reproducing kernels of harmonic Bergman–Besov spaces. We consider a class of integral operators related to harmonic Bergman projection and determine precisely when they are bounded on \(L^\infty _\alpha \). We define projections from \(L^\infty _\alpha \) to \(b_\alpha \) and as a consequence obtain integral representations. We solve the Gleason problem and provide atomic decomposition for all \(b_\alpha , \alpha \in {\mathbb {R}}\). Finally we give an oscillatory characterization of \(b_\alpha \) when \(\alpha >-1\).