We prove bounds for the truncated directional Hilbert transform in $L^{p}(\mathbb{R}^{2})$ for any $1<p<\infty$ under a combination of a Lipschitz assumption and a lacunarity assumption. It is known that a lacunarity assumption alone is not sufficient to yield boundedness for $p=2$, and it is a major question in the field whether a Lipschitz assumption alone suffices, at least for some $p$.

We prove the analog of Cramér’s short intervals theorem for primes in arithmetic progressions and prime ideals, under the relevant Riemann hypothesis. Both results are uniform in the data of the underlying structure. Our approach is based mainly on the inertia property of the counting functions of primes and prime ideals.

Let $n\geqslant C$ for a large universal constant $C>0$ and let $B$ be a convex body in $\mathbb{R}^{n}$ such that for any $(x_{1},x_{2},\ldots ,x_{n})\in B$, any choice of signs $\unicode[STIX]{x1D700}_{1},\unicode[STIX]{x1D700}_{2},\ldots ,\unicode[STIX]{x1D700}_{n}\in \{-1,1\}$ and for any permutation $\unicode[STIX]{x1D70E}$ on $n$ elements, we have $(\unicode[STIX]{x1D700}_{1}x_{\unicode[STIX]{x1D70E}(1)},\unicode[STIX]{x1D700}_{2}x_{\unicode[STIX]{x1D70E}(2)},\ldots ,\unicode[STIX]{x1D700}_{n}x_{\unicode[STIX]{x1D70E}(n)})\in B$. We show that if $B$ is not a cube, then $B$ can be illuminated by strictly less than $2^{n}$ sources of light. This confirms the Hadwiger–Gohberg–Markus illumination conjecture for unit balls of $1$-symmetric norms in $\mathbb{R}^{n}$ for all sufficiently large $n$.

Let $X$ be a vector space and let $\unicode[STIX]{x1D711}:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$ be an extended real-valued function. For every function $f:X\rightarrow \mathbb{R}\cup \{-\infty ,+\infty \}$, let us define the $\unicode[STIX]{x1D711}$-envelope of $f$ by

$$\begin{eqnarray}f^{\unicode[STIX]{x1D711}}(x)=\sup _{y\in X}\unicode[STIX]{x1D711}(x-y)\begin{array}{@{}c@{}}-\\ \cdot \end{array}f(y),\end{eqnarray}$$

$\begin{array}{@{}c@{}}-\\ \\ \\ \cdot \end{array}$

$\mathbb{R}\cup \{-\infty ,+\infty \}$

$f\mapsto f^{\unicode[STIX]{x1D711}}$

$\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D711}=(1/p\unicode[STIX]{x1D706})\Vert \cdot \Vert ^{p}$

$\unicode[STIX]{x1D706}>0$

$p\geqslant 1$

$\unicode[STIX]{x1D706}$

$p$

$\unicode[STIX]{x1D711}$

$-\unicode[STIX]{x1D711}$

$\unicode[STIX]{x1D711}$

$X$

$\unicode[STIX]{x1D6EC}$

$C\subset X$

$\unicode[STIX]{x1D6EC}$

$C$

$C^{\unicode[STIX]{x1D6EC}}=\bigcap _{x\in C}(x+\unicode[STIX]{x1D6EC})$

$C\mapsto C^{\unicode[STIX]{x1D6EC}}$

$\unicode[STIX]{x1D711}$

The paper is devoted to the study of Fefferman–Stein inequalities for stochastic integrals. If $X$ is a martingale, $Y$ is the stochastic integral, with respect to $X$, of some predictable process taking values in $[-1,1]$, then for any weight $W$ belonging to the class $A_{1}$ we have the estimates $\Vert Y_{\infty }\Vert _{L^{p}(W)}\leqslant 8pp^{\prime }[W]_{A_{1}}\Vert X_{\infty }\Vert _{L^{p}(W)},$$1<p<\infty ,$ and $\Vert Y_{\infty }\Vert _{L^{1,\infty }(W)}\leqslant c[W]_{A_{1}}(1+\log [W]_{A_{1}})\Vert X_{\infty }\Vert _{L^{1}(W)}.$ The proofs rest on the Bellman function method: the inequalities are deduced from the existence of certain special functions, enjoying appropriate majorization and concavity. As an application, related statements for Haar multipliers are indicated. The above estimates can be regarded as probabilistic counterparts of the recent results of Lerner, Ombrosi and Pérez concerning singular integral operators.

Let $p,q$ be primes such that $q|p-1$ and set $\unicode[STIX]{x1D6F7}=C_{p}\rtimes C_{q}$, $G=\unicode[STIX]{x1D6F7}\times C_{\infty }^{n}$ and $\unicode[STIX]{x1D6EC}=\mathbf{Z}[G]$, the integral group ring of $G$. By means of a fibre square decomposition, we show that stably free modules over $\unicode[STIX]{x1D6EC}$ are necessarily free.

We obtain explicit inversion formulas for the Radon-like transform that assigns to a function on the unit sphere the integrals of that function over hemispheres lying in lower-dimensional central cross-sections. The results are applied to the determination of star bodies from the volumes of their central half-sections.

We consider the minimization of Dirichlet eigenvalues $\unicode[STIX]{x1D706}_{k}$, $k\in \mathbb{N}$, of the Laplacian on cuboids of unit measure in $\mathbb{R}^{3}$. We prove that any sequence of optimal cuboids in $\mathbb{R}^{3}$ converges to a cube of unit measure in the sense of Hausdorff as $k\rightarrow \infty$. We also obtain an upper bound for that rate of convergence.

We prove an asymptotic formula for squarefree numbers in arithmetic progressions, improving previous results by Prachar and Hooley. As a consequence we improve a lower bound of Heath-Brown for the least squarefree number in an arithmetic progression.

We consider the notion of a free resolution. In general, a free resolution can be of any length depending on the group ring under investigation. The metacyclic groups $G(pq)$ however admit periodic resolutions. In the particular case of $G(21)$ it is possible to achieve a fully diagonalized resolution. In order to achieve a diagonal resolution, we obtain a complete list of indecomposable modules over $\unicode[STIX]{x1D6EC}$. Such a list aids the decomposition of the augmentation ideal (the first syzygy) into a direct sum of indecomposable modules. Therefore, we are able to achieve a diagonalized map here. From this point it is possible to decompose all of the remaining syzygies in terms of indecomposable modules, leaving a diagonal resolution.

For a non-autonomous dynamics defined by a sequence of matrices, we consider the notion of a non-uniform exponential trichotomy for an arbitrary growth rate (this means that there may exist contracting, expanding and neutral directions with an arbitrary fixed growth rate). The purpose of our work is two-fold: to use a regularity coefficient in order to show that these trichotomies occur naturally and to provide several alternative characterizations of those for which the non-uniform part is arbitrarily small. This includes characterizations in terms of the growth rate of volumes and of the Lyapunov exponents of the dynamics and its adjoint. We also obtain sharp lower and upper bounds for the regularity coefficient.

Let $E$ be a finite-dimensional normed space and $\unicode[STIX]{x1D6FA}$ a non-empty convex open set in $E$. We show that the Lipschitz-free space of $\unicode[STIX]{x1D6FA}$ is canonically isometric to the quotient of $L^{1}(\unicode[STIX]{x1D6FA},E)$ by the subspace consisting of vector fields with zero divergence in the sense of distributions on $E$.

We improve a recent result of B. Hanson [Estimates for character sums with various convolutions. Preprint, 2015, arXiv:1509.04354] on multiplicative character sums with expressions of the type $a+b+cd$ and variables $a,b,c,d$ from four distinct sets of a finite field. We also consider similar sums with $a+b(c+d)$. Our new bounds rely on some recent advances in additive combinatorics.

For every integer $k\geqslant 3$ we construct a $k$-gonal curve $C$ along with a very ample divisor of degree $2g+k-1$ (where $g$ is the genus of $C$) to which the vanishing statement from the Green–Lazarsfeld gonality conjecture does not apply.

Let $h(d)$ be the class number of indefinite binary quadratic forms of discriminant $d$ and let $\unicode[STIX]{x1D700}_{d}$ be the corresponding fundamental unit. In this paper, we obtain an asymptotic formula for the $k$th moment of $h(d)$ over positive discriminants $d$ with $\unicode[STIX]{x1D700}_{d}\leqslant x$, uniformly for real numbers $k$ in the range $0<k\leqslant (\log x)^{1-o(1)}$. This improves upon the work of Raulf, who obtained such an asymptotic for a fixed positive integer $k$. We also investigate the distribution of large values of $h(d)$ when the discriminants $d$ are ordered according to the size of their fundamental units $\unicode[STIX]{x1D700}_{d}$. In particular, we show that the tail of this distribution has the same shape as that of class numbers of imaginary quadratic fields ordered by the size of their discriminants. As an application of these results, we prove that there are many positive discriminants $d$ with class number $h(d)\geqslant (e^{\unicode[STIX]{x1D6FE}}/3+o(1))\cdot \unicode[STIX]{x1D700}_{d}(\log \log \unicode[STIX]{x1D700}_{d})/\log \,\unicode[STIX]{x1D700}_{d}$, a bound that we believe is best possible. We also obtain an upper bound for $h(d)$ that is twice as large, assuming the generalized Riemann hypothesis.

This paper is motivated by Davenport’s problem and the subsequent work regarding badly approximable points in submanifolds of a Euclidean space. We study the problem in the area of twisted Diophantine approximation and present two different approaches. The first approach shows that, under a certain restriction, any countable intersection of the sets of weighted badly approximable points on any non-degenerate ${\mathcal{C}}^{1}$ submanifold of $\mathbb{R}^{n}$ has full dimension. In the second approach, we introduce the property of isotropically winning and show that the sets of weighted badly approximable points are isotropically winning under the same restriction as above.

We introduce a new arithmetic function $a(n)$ defined to be the number of random multiplications by residues modulo $n$ before the running product is congruent to zero modulo $n$. We give several formulas for computing the values of this function and analyze its asymptotic behavior. We find that it is closely related to $P_{1}(n)$, the largest prime divisor of $n$. In particular, $a(n)$ and $P_{1}(n)$ have the same average order asymptotically. Furthermore, the difference between the functions $a(n)$ and $P_{1}(n)$ is $o(1)$ as $n$ tends to infinity on a set with density approximately $0.623$. On the other hand, however, we see that (except on a set of density zero) the difference between $a(n)$ and $P_{1}(n)$ tends to infinity on the integers outside this set. Finally, we consider the asymptotic behavior of the difference between these two functions and find that $\sum _{n\leqslant x}(a(n)-P_{1}(n))\sim (1-\unicode[STIX]{x1D70B}/4)\sum _{n\leqslant x}P_{2}(n)$, where $P_{2}(n)$ is the second largest divisor of $n$.

Recently, an analogue over $\mathbb{F}_{q}[T]$ of Landau’s theorem on sums of two squares was considered by Bary-Soroker, Smilansky and Wolf. They counted the number of monic polynomials in $\mathbb{F}_{q}[T]$ of degree $n$ of the form $A^{2}+TB^{2}$, which we denote by $B(n,q)$. They studied $B(n,q)$ in two limits: fixed $n$ and large $q$; and fixed $q$ and large $n$. We generalize their result to the most general limit $q^{n}\rightarrow \infty$. More precisely, we prove

$$\begin{eqnarray}B(n,q)\sim K_{q}\cdot \binom{n-\frac{1}{2}}{n}\cdot q^{n},\quad q^{n}\rightarrow \infty ,\end{eqnarray}$$

$K_{q}=1+O(1/q)$

We prove that if $k$ and $\ell$ are sufficiently large, then all the zeros of the weight $k+\ell$ cusp form $E_{k}(z)E_{\ell }(z)-E_{k+\ell }(z)$ in the standard fundamental domain lie on the boundary. We, moreover, find formulas for the number of zeros on the bottom arc with $|z|=1$, and those on the sides with $Re(z)=\pm 1/2$. One important ingredient of the proof is an approximation of the Eisenstein series in terms of the Jacobi theta function.

Answering a question of Füredi and Loeb [On the best constant for the Besicovitch covering theorem. Proc. Amer. Math. Soc.121(4) (1994), 1063–1073], we show that the maximum number of pairwise intersecting homothets of a $d$-dimensional centrally symmetric convex body $K$, none of which contains the center of another in its interior, is at most $O(3^{d}d\log d)$. If $K$ is not necessarily centrally symmetric and the role of its center is played by its centroid, then the above bound can be replaced by $O(3^{d}\binom{2d}{d}d\log d)$. We establish analogous results for the case where the center is defined as an arbitrary point in the interior of $K$. We also show that, in the latter case, one can always find families of at least $\unicode[STIX]{x1D6FA}((2/\sqrt{3})^{d})$ translates of $K$ with the above property.