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 $I_{s,k,r}(X)$ denote the number of integral solutions of the modified Vinogradov system of equations

$$\begin{eqnarray}x_{1}^{j}+\cdots +x_{s}^{j}=y_{1}^{j}+\cdots +y_{s}^{j}\quad (1\leqslant j\leqslant k,\;j\neq r),\end{eqnarray}$$

$1\leqslant x_{i},y_{i}\leqslant X\;(1\leqslant i\leqslant s)$

$I_{s,k,r}(X)$

$1\leqslant r\leqslant k-1$

$s,k\in \mathbb{N}$

$k\geqslant 3$

$1\leqslant s\leqslant (k^{2}-1)/2$

$I_{s,k,1}(X)\ll X^{s+\unicode[STIX]{x1D700}}$

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 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 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.

We consider an arithmetic function defined independently by John G. Thompson and Greg Simay, with particular attention to its mean value, its maximal size, and the analytic nature of its Dirichlet series generating function.

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.

There are several formulas in classical prime number theory that are said to be “equivalent” to the Prime Number Theorem. For Beurling generalized numbers, not all such implications hold unconditionally. Here we investigate conditions under which the Beurling version of these relations do or do not hold.

Combinatorial discrepancy is a complexity measure of a collection of sets which quantifies how well the sets in the collection can be simultaneously balanced. More precisely, we are given an $n$ -point set $P$ , and a collection ${\mathcal{F}}=\{F_{1},\ldots ,F_{m}\}$ of subsets of $P$ , and our goal is color $P$ with two colors, red and blue, so that the maximum over the $F_{i}$ of the absolute difference between the number of red elements and the number of blue elements (the discrepancy) is minimized. Combinatorial discrepancy has many applications in mathematics and computer science, including constructions of uniformly distributed point sets, and lower bounds for data structures and private data analysis algorithms. We investigate the combinatorial discrepancy of geometrically defined systems, in which $P$ is an $n$ -point set in $d$ -dimensional space, and ${\mathcal{F}}$ is the collection of subsets of $P$ induced by dilations and translations of a fixed convex polytope $B$ . Such set systems include systems of sets induced by axis-aligned boxes, whose discrepancy is the subject of the well-known Tusnády problem. We prove new discrepancy upper and lower bounds for such set systems by extending the approach based on factorization norms previously used by the author, Matoušek, and Talwar. We also outline applications of our results to geometric discrepancy, data structure lower bounds, and differential privacy.

The discrepancy function measures the deviation of the empirical distribution of a point set in $[0,1]^{d}$ from the uniform distribution. In this paper, we study the classical discrepancy function with respect to the bounded mean oscillation and exponential Orlicz norms, as well as Sobolev, Besov and Triebel–Lizorkin norms with dominating mixed smoothness. We give sharp bounds for the discrepancy function under such norms with respect to infinite sequences.

Introduced in Schmidt and Summerer [Parametric geometry of numbers and applications. Acta Arith.140 (2009), 67–91], approximation exponents $\text{}\underline{\unicode[STIX]{x1D711}}_{i},\overline{\unicode[STIX]{x1D711}}_{i}$ , $(i=1,2,3)$ , attached to points $\boldsymbol{\unicode[STIX]{x1D709}}=(\unicode[STIX]{x1D709}_{1},\unicode[STIX]{x1D709}_{2})$ in $\mathbb{R}^{2}$ , give information on Diophantine approximation properties of these points. Laurent [Exponents of Diophantine approximation in dimension two. Canad. J. Math.61 (2009), 165–189] had described all possible quadruples $(\text{}\underline{\unicode[STIX]{x1D711}}_{1},\overline{\unicode[STIX]{x1D711}}_{1},\text{}\underline{\unicode[STIX]{x1D711}}_{3},\overline{\unicode[STIX]{x1D711}}_{3})$ arising in this way. Our emphasis here will be on $\text{}\underline{\unicode[STIX]{x1D711}}_{2},\overline{\unicode[STIX]{x1D711}}_{2}$ and the construction of suitable “ $3$ -systems”.

The multiplicative group generated by a certain sequence of rationals is determined, settling a 30-year conjecture.

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

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

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

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.

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.

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 $E$ .