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In this paper we study some cube packing problems. In particular we are interested in compact subsets of $\mathbb{R}^n,n\geq 2$, which contain boundaries of cubes with all side lengths in $(0,1)$. We show here that such sets must have lower box dimension at least $n-0.5$ and we will also provide sharp examples. We also show here that such sets must be large in general in a precise sense which is also introduced in this paper., 9 pages. arXiv admin note: text overlap with arXiv:1711.06533

Starting with a paper of Jacobson from the 1960s, many authors became interested in characterizing all algebraic number fields in which each integer is the sum of pairwise distinct units. Although there exist many partial results for number fields of low degree, a full characterization of these number fields is still not available. Narkiewicz and Jarden posed an analogous question for sums of units that are not necessarily distinct. In this paper we propose a generalization of these problems. In particular, for a given rational integer n we consider the following problem. Characterize all number fields for which every integer is a linear combination of finitely many units e i in a way that the coefficients a i ∈ℕ are bounded by n . The paper gives several partial results on this problem. In our proofs we exploit the fact that these representations are related to symmetric beta expansions with respect to Pisot bases.

The contact number of a packing of finitely many balls in Euclidean $d$-space is the number of touching pairs of balls in the packing. A prominent subfamily of sphere packings is formed by the so-called totally separable sphere packings: here, a packing of balls in Euclidean $d$-space is called totally separable if any two balls can be separated by a hyperplane such that it is disjoint from the interior of each ball in the packing. Bezdek, Szalkai and Szalkai (Discrete Math. 339(2): 668-676, 2016) upper bounded the contact numbers of totally separable packings of $n$ unit balls in Euclidean $d$-space in terms of $n$ and $d$. In this paper we improve their upper bound and extend that new upper bound to the so-called locally separable packings of unit balls. We call a packing of unit balls a locally separable packing if each unit ball of the packing together with the unit balls that are tangent to it form a totally separable packing. In the plane, we prove a crystallization result by characterizing all locally separable packings of $n$ unit disks having maximum contact number.

Let $\mathbb{F}$ be a fixed field and let $X$ be a simplicial complex on the vertex set $V$. The Leray number $L(X;\mathbb{F})$ is the minimal $d$ such that for all $i \geq d$ and $S \subset V$, the induced complex $X[S]$ satisfies $\tilde{H}_i(X[S];\mathbb{F})=0$. Leray numbers play a role in formulating and proving topological Helly type theorems. For two complexes $X,Y$ on the same vertex set $V$, define the relative Leray number $L_Y(X;\mathbb{F})$ as the minimal $d$ such that $\tilde{H}_i(X[V \setminus \sigma];\mathbb{F})=0$ for all $i \geq d$ and $\sigma \in Y$. In this paper we extend some results on Leray numbers to the relative setting. In particular, we give an alternative characterization of $L_Y(X;\mathbb{F})$ in terms of links, and prove a relative version of the topological colorful Helly theorem. Our main tools are the Zeeman spectral sequence and a Mayer-Vietoris type spectral sequence.

Let $K=\mathbb{Q}(\sqrt[n]{a})$ be an extension of degree $n$ of the field $\Q$ of rational numbers, where the integer $a$ is such that for each prime $p$ dividing $n$ either $p\nmid a$ or the highest power of $p$ dividing $a$ is coprime to $p$; this condition is clearly satisfied when $a, n$ are coprime or $a$ is squarefree. The present paper gives explicit construction of an integral basis of $K$ along with applications. This construction of an integral basis of $K$ extends a result proved in [J. Number Theory, {173} (2017), 129-146] regarding periodicity of integral bases of $\mathbb{Q}(\sqrt[n]{a})$ when $a$ is squarefree.

A conjecture of Leader, Russell and Walters in Euclidean Ramsey theory says that a finite set is Ramsey if and only if it is congruent to a subset of a set whose symmetry group acts transitively. As they have shown the ``if" direction of their conjecture follows if all finite groups have a Hales--Jewett type property. In this paper, we show that this property is satisfied in the case of finite solvable groups. Our result can be used to recover the work of Křiž in Euclidean Ramsey theory.

Let $N$ balls of the same radius be given in a $d$-dimensional real normed vector space, i.e., in a Minkowski $d$-space. Then apply a uniform contraction to the centers of the $N$ balls without changing the common radius. Here a uniform contraction is a contraction where all the pairwise distances in the first set of centers are larger than all the pairwise distances in the second set of centers. The main results of this paper state that a uniform contraction of the centers does not increase (resp., decrease) the volume of the union (resp., intersection) of $N$ balls in Minkowski $d$-space, provided that $N\geq 2^d$ (resp., $N\geq 3^d$ and the unit ball of the Minkowski $d$-space is a generating set). Some improvements are presented in Euclidean spaces.

In this paper we prove transference inequalities for regular and uniform Diophantine exponents in the weighted setting. Our results generalize the corresponding inequalities that exist in the `non-weighted' case.

We study how small is the set of critical values of the distance function from a compact (resp. closed) set in the plane or in a connected complete two-dimensional Riemannian manifold. We show that for a compact set, the set of critical values is compact and Lebesgue null (which is a known result) and that it has "locally" (away from 0) bounded sum of square roots of lengths of gaps (components of the complement). In the planar case, these conditions of local smallness are shown to be optimal. These results improve and generalize those of Fu (1985) and of our earlier paper from 2012. We also find an optimal condition for the smallness of the whole set of critical values of a planar compact set.

In this paper, we study the generalized Lebesgue-Nagell equation \[ x^2+7^{2k+1}=y^n. \] This is the last case of equations of the form $x^2+q^{2k+1}=y^n$ with $k\geq0$ and $q>0$ where $\mathbb{Q}(\sqrt{-q})$ has class number one. Our proof is based on the modular method and the symplectic argument.

In this paper we study the number of rational points on curves in an ensemble of abelian covers of the projective line: Let $\ell$ be a prime, $q$ a prime power and consider the ensemble $\mathcal{H}_{g,\ell}$ of $\ell$-cyclic covers of $\mathbb{P}^1_{\mathbb{F}_q}$ of genus $g$. We assume that $q\not\equiv 0,1\mod \ell$. If $2g+2\ell-2\not\equiv0\mod (\ell-1){\rm ord}_\ell(q)$, then $\mathcal{H}_{g,\ell}$ is empty. Otherwise, the number of rational points on a random curve in $\mathcal{H}_{g,\ell}$ distributes as $\sum_{i=1}^{q+1} X_i$ as $g\to \infty$, where $X_1,\ldots, X_{q+1}$ are i.i.d.\ random variables taking the values $0$ and $\ell$ with probabilities $\frac{\ell-1}{\ell}$ and $\frac{1}{\ell}$, respectively. The novelty of our result is that it works in the absence of a primitive $\ell$-th-root of unity, the presence of which was crucial in previous studies., We have corrected a mistake in the main theorem pointed out by Will Sawin

We consider point distributions in compact connected two-point homogeneous spaces (Riemannian symmetric spaces of rank one). All such spaces are known: they are the spheres in the Euclidean spaces, the real, complex and quaternionic projective spaces and the octonionic projective plane. For all such spaces the best possible bounds for the quadratic discrepancies and sums of pairwise distances are obtained in the paper (Theorems 2.1 and 2.2). Distributions of points of -designs meet the best possible bounds for quadratic discrepancies and sums of pairwise distances (Corollary 2.1). Our approach is based on the Fourier analysis on two-point homogeneous spaces and explicit spherical function expansions for discrepancies and sums of distances (Theorems 4.1 and 4.2).

The paper is concerned with the Steklov eigenvalue problem on cuboids of arbitrary dimension. We prove a two-term asymptotic formula for the counting function of Steklov eigenvalues on cuboids in dimension d greater or equal to 3. Apart from the standard Weyl term, we calculate explicitly the second term in the asymptotics, capturing the contribution of the (d-2)-dimensional facets of a cuboid. Our approach is based on lattice counting techniques. While this strategy is similar to the one used for the Dirichlet Laplacian, the Steklov case carries additional complications. In particular, it is not clear how to establish directly the completeness of the system of Steklov eigenfunctions admitting separation of variables. We prove this result using a family of auxiliary Robin boundary value problems. Moreover, the correspondence between the Steklov eigenvalues and lattice points is not exact, and hence more delicate analysis is required to obtain spectral asymptotics. Some other related results are presented, such as an isoperimetric inequality for the first Steklov eigenvalue, a concentration property of high frequency Steklov eigenfunctions and applications to spectral determination of cuboids., Comment: 31 pages, 3 figures

In this paper we provide a unified approach, based on methods of descriptive set theory, for proving some classical selection theorems. Among them is the zero-dimensional Michael selection theorem, the Kuratowski–Ryll-Nardzewski selection theorem, as well as a known selection theorem for hyperspaces.

For the superelliptic curves of the form with , , a prime and for , we show that Here denotes the interval , where is the least prime greater than or equal to . Bennett and Siksek obtained a similar bound for in a recent paper.

In the paper, we provide an effective method for the Lipschitz equivalence of two-branch Cantor sets and three-branch Cantor sets by studying the irreducibility of polynomials. We also find that any two Cantor sets are Lipschitz equivalent if and only if their contraction vectors are equivalent provided one of the contraction vectors is homogeneous., Comment: 12 pages

The paper is devoted to the study of Fefferman–Stein inequalities for stochastic integrals. If is a martingale, is the stochastic integral, with respect to , of some predictable process taking values in , then for any weight belonging to the class we have the estimates and 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 Perez concerning singular integral operators.

Define to be the largest cardinality of a set that does not contain four elements in arithmetic progression. In 1998, Gowers proved that for some absolute constant . In 2005, the authors improved this to In this paper we further improve this to which appears to be the limit of our methods.

In this paper we show that sieve methods used previously to investigate primes in short intervals and corresponding Goldbach-type problems can be modified to obtain results on primes in Beatty sequences in short intervals.

This paper presents a new result concerning the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve over an arbitrary number field K with a single point of order two that does not have a cyclic 4-isogeny defined over its two-division field. We prove that at least half of all the quadratic twists of such an elliptic curve have arbitrarily large 2-Selmer rank, showing that the distribution of 2-Selmer ranks in the quadratic twist family of such an elliptic curve differs from the distribution of 2-Selmer ranks in the quadratic twist family of an elliptic curve having either no rational two-torsion or full rational two-torsion., Comment: arXiv admin note: text overlap with arXiv:1203.1030

An integer $d$ is called a jumping champion for a given $x$ if $d$ is the most common gap between consecutive primes up to $x$. Occasionally several gaps are equally common. Hence, there can be more than one jumping champion for the same $x$. For the $n$th prime $p_{n}$, the $n$th primorial $p_{n}^{\sharp}$ is defined as the product of the first $n$ primes. In 1999, Odlyzko, Rubinstein and Wolf provided convincing heuristics and empirical evidence for the truth of the hypothesis that the jumping champions greater than 1 are 4 and the primorials $p_{1}^{\sharp}, p_{2}^{\sharp}, p_{3}^{\sharp}, p_{4}^{\sharp}, p_{5}^{\sharp}, ...$, that is, $2, 6, 30, 210, 2310, ....$ In this paper, we prove that an appropriate form of the Hardy-Littlewood prime $k$-tuple conjecture for prime pairs and prime triples implies that all sufficiently large jumping champions are primorials and that all sufficiently large primorials are jumping champions over a long range of $x$., 19 pages, 1 table

Asymptotic analysis of the Hele-Shaw flow with a small moving obstacle is performed. The method of solution utilises the uniform asymptotic formulas for Green's and Neumann functions recently obtained by V. Maz'ya and A. Movchan. Theoretical results of the paper are illustrated by the numerical simulations., 29 pages, 14 figures

1. The purpose of this paper is to give simple proofs for some recent versions of Linnik's large sieve, and some applications.The first theme of the large sieve is that an arbitrary set of Z integers in an interval of length N must be well distributed among most of the residue classes modulo p, for most small primes p, unless Z is small compared with N. Following improvements on Linnik's original result [1] by Renyi [2] and by Roth [3], Bombieri [4] recently proved the following inequality: Denote by Z(a, p) the number of integers in the set which are congruent to a modulo p.