We generalize to higher dimensions the Bavard–Ghys construction of the hyperbolic metric on the space of polygons with fixed directions of edges. The space of convex $d$-dimensional polyhedra with fixed directions of facet normals has a decomposition into type cones that correspond to different combinatorial types of polyhedra. This decomposition is a subfan of the secondary fan of a vector configuration and can be analyzed with the help of Gale diagrams. We construct a family of quadratic forms on each of the type cones using the theory of mixed volumes. The Alexandrov–Fenchel inequalities ensure that these forms have exactly one positive eigenvalue. This introduces a piecewise hyperbolic structure on the space of similarity classes of polyhedra with fixed directions of facet normals. We show that some of the dihedral angles on the boundary of the resulting cone-manifold are equal to $\unicode[STIX]{x1D70B}/2$.

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 say that a planar set $A$ has the Kakeya property if there exist two different positions of $A$ such that $A$ can be continuously moved from the first position to the second within a set of arbitrarily small area. We prove that if $A$ is closed and has the Kakeya property, then the union of the non-trivial connected components of $A$ can be covered by a null set which is either the union of parallel lines or the union of concentric circles. In particular, if $A$ is closed, connected and has the Kakeya property, then $A$ can be covered by a line or a circle.

Building on recent work of Bhargava, Elkies and Schnidman and of Kriz and Li, we produce infinitely many smooth cubic surfaces defined over the field of rational numbers that contain rational points.

Let $S=\{q_{1},\ldots ,q_{s}\}$ be a finite, non-empty set of distinct prime numbers. For a non-zero integer $m$, write $m=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}M$, where $r_{1},\ldots ,r_{s}$ are non-negative integers and $M$ is an integer relatively prime to $q_{1}\cdots q_{s}$. We define the $S$-part $[m]_{S}$ of $m$ by $[m]_{S}:=q_{1}^{r_{1}}\cdots q_{s}^{r_{s}}$. Let $(u_{n})_{n\geqslant 0}$ be a linear recurrence sequence of integers. Under certain necessary conditions, we establish that for every $\unicode[STIX]{x1D700}>0$, there exists an integer $n_{0}$ such that $[u_{n}]_{S}\leqslant |u_{n}|^{\unicode[STIX]{x1D700}}$ holds for $n>n_{0}$. Our proof is ineffective in the sense that it does not give an explicit value for $n_{0}$. Under various assumptions on $(u_{n})_{n\geqslant 0}$, we also give effective, but weaker, upper bounds for $[u_{n}]_{S}$ of the form $|u_{n}|^{1-c}$, where $c$ is positive and depends only on $(u_{n})_{n\geqslant 0}$ and $S$.

Assuming the Riemann Hypothesis, Soundararajan [Ann. of Math. (2) 170 (2009), 981–993] showed that $\int _{0}^{T}|\unicode[STIX]{x1D701}(1/2+\text{i}t)|^{2k}\ll T(\log T)^{k^{2}+\unicode[STIX]{x1D716}}$. His method was used by Chandee [Q. J. Math.62 (2011), 545–572] to obtain upper bounds for shifted moments of the Riemann Zeta function. Building on these ideas of Chandee and Soundararajan, we obtain, conditionally, upper bounds for shifted moments of Dirichlet $L$-functions which allow us to derive upper bounds for moments of theta functions.

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.

Let $\unicode[STIX]{x1D703}$ be an arithmetic function and let ${\mathcal{B}}$ be the set of positive integers $n=p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{k}^{\unicode[STIX]{x1D6FC}_{k}}$ which satisfy $p_{j+1}\leqslant \unicode[STIX]{x1D703}(p_{1}^{\unicode[STIX]{x1D6FC}_{1}}\cdots p_{j}^{\unicode[STIX]{x1D6FC}_{j}})$ for $0\leqslant j<k$. We show that ${\mathcal{B}}$ has a natural density, provide a criterion to determine whether this density is positive, and give various estimates for the counting function of ${\mathcal{B}}$. When $\unicode[STIX]{x1D703}(n)/n$ is non-decreasing, the set ${\mathcal{B}}$ coincides with the set of integers $n$ whose divisors $1=d_{1}<d_{2}<\cdots <d_{\unicode[STIX]{x1D70F}(n)}=n$ satisfy $d_{j+1}\leqslant \unicode[STIX]{x1D703}(d_{j})$ for $1\leqslant j<\unicode[STIX]{x1D70F}(n)$.

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.

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.

Euler noted the relation $6^{3}\,=\,3^{3}+4^{3}+5^{3}$ and asked for other instances of cubes that are sums of consecutive cubes. Similar problems have been studied by Cunningham, Catalan, Gennochi, Lucas, Pagliani, Cassels, Uchiyama, Stroeker and Zhongfeng Zhang. In particular, Stroeker determined all squares that can be written as a sum of at most 50 consecutive cubes. We generalize Stroeker’s work by determining all perfect powers that are sums of at most 50 consecutive cubes. Our methods include descent, linear forms in two logarithms and Frey–Hellegouarch curves.

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.

We show that smooth-supported multiplicative functions $f$ are well distributed in arithmetic progressions $a_{1}a_{2}^{-1}\;(\text{mod}~q)$ on average over moduli $q\leqslant x^{3/5-\unicode[STIX]{x1D700}}$ with $(q,a_{1}a_{2})=1$.

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 study the Hausdorff measure and dimension of the set of intrinsically simultaneously $\unicode[STIX]{x1D713}$-approximable points on a curve, surface, etc, given as a graph of integer polynomials. We obtain complete answers to these questions for algebraically “nice” manifolds. This generalizes earlier work done in the case of curves.

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.

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

We present a first example of a flag vector of a polyhedral sphere that is not the flag vector of any polytope. Namely, there is a unique $3$-sphere with the parameters $(f_{0},f_{1},f_{2},f_{3};f_{02})=(12,40,40,12;120)$, but this sphere is not realizable by a convex $4$-polytope. The $3$-sphere, which is $2$-simple and $2$-simplicial, was found by Werner [Linear constraints on face numbers of polytopes. PhD Thesis, TU Berlin, Germany, 2009]; we present results of a computer enumeration which imply that the sphere with these parameters is unique. We prove that it is non-polytopal in two ways: first, we show that it has no oriented matroid, and thus it is not realizable; this proof was found by computer, but can be verified by hand. The second proof is again a computer-based oriented matroid proof and shows that for exactly one of the facets this sphere does not even have a diagram based on this facet. Using the non-polytopality, we finally prove that the sphere is not even embeddable as a polytopal complex.

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.